from abc import ABC, abstractmethod
from math import exp, log
from typing import TYPE_CHECKING
+
import numpy as np
-from mici.errors import IntegratorError, AdaptationError
-from mici.matrices import PositiveDiagonalMatrix, DensePositiveDefiniteMatrix
+
+from mici.errors import AdaptationError, IntegratorError
+from mici.matrices import DensePositiveDefiniteMatrix, PositiveDiagonalMatrix
if TYPE_CHECKING:
- from typing import Collection, Optional, Iterable, Union
+ from typing import Collection, Iterable, Optional, Union
+
from numpy.random import Generator
from numpy.typing import ArrayLike
+
from mici.integrators import Integrator
from mici.states import ChainState
from mici.systems import System
from mici.transitions import Transition
- from mici.types import (
- AdaptationStatisticFunction,
- AdapterState,
- ReducerFunction,
- TransitionStatistics,
- )
+ from mici.types import (AdaptationStatisticFunction, AdapterState,
+ ReducerFunction, TransitionStatistics)
[docs]class Adapter(ABC):
@@ -213,7 +213,7 @@ Source code for mici.adapters
return exp(sum(x for x in log_step_sizes) / len(log_step_sizes))
-[docs]def min_log_step_size_reducer(log_step_sizes: Iterable[float]) -> float:
+[docs]def min_log_step_size_reducer(log_step_sizes: Collection[float]) -> float:
"""Compute minimum of step sizes from their logs.
Args:
@@ -226,7 +226,7 @@ Source code for mici.adapters
[docs]def default_adapt_stat_func(stats: TransitionStatistics) -> float:
- """Function to extract default acceptance statistic used for step-size adaptation.
+ """Function to extract default statistic used for step-size adaptation.
Args:
stats: Dictionary of transition statistics.
@@ -241,18 +241,18 @@ Source code for mici.adapters
"""Dual averaging integrator step size adapter.
Implementation of the dual algorithm step size adaptation algorithm described in
- [1], a modified version of the stochastic optimisation scheme of [2]. By default the
- adaptation is performed to control the `accept_prob` statistic of an integration
- transition to be close to a target value but the statistic adapted on can be altered
- by changing the `adapt_stat_func`.
-
+ Hoffman and Gelman (2014), a modified version of the stochastic optimisation scheme
+ of Nesterov (2009). By default the adaptation is performed to control the
+ :code:`accept_stat` statistic of an integration transition to be close to a target
+ value but the statistic adapted on can be altered by changing the
+ :code:`adapt_stat_func`.
References:
- 1. Hoffman, M.D. and Gelman, A., 2014. The No-U-turn sampler: adaptively setting
+ 1. Hoffman, M.D. and Gelman, A. (2014). The No-U-turn sampler: adaptively setting
path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research,
15(1), pp.1593-1623.
- 2. Nesterov, Y., 2009. Primal-dual subgradient methods for convex problems.
+ 2. Nesterov, Y. (2009). Primal-dual subgradient methods for convex problems.
Mathematical programming 120(1), pp.221-259.
"""
@@ -275,19 +275,19 @@ Source code for mici.adapters
being controlled during adaptation.
adapt_stat_func: Function which given a dictionary of transition statistics
outputs the value of the statistic to control during adaptation. By
- default this is set to a function which simply selects the 'accept_stat'
- value in the statistics dictionary.
+ default this is set to a function which simply selects the
+ :code:'accept_stat' value in the statistics dictionary.
log_step_size_reg_target: Value to regularize the controlled output
- (logarithm of the integrator step size) towards. If `None` set to
- `log(10 * init_step_size)` where `init_step_size` is the initial
- 'reasonable' step size found by a coarse search as recommended in
- Hoffman and Gelman (2014). This has the effect of giving the dual
+ (logarithm of the integrator step size) towards. If :code:`None` set to
+ :code:`log(10 * init_step_size)` where :code:`init_step_size` is the
+ initial 'reasonable' step size found by a coarse search as recommended
+ in Hoffman and Gelman (2014). This has the effect of giving the dual
averaging algorithm a tendency towards testing step sizes larger than
the initial value, with typically integrating with a larger step size
having a lower computational cost.
log_step_size_reg_coefficient: Coefficient controlling amount of
regularisation of controlled output (logarithm of the integrator step
- size) towards `log_step_size_reg_target`. Defaults to 0.05 as
+ size) towards :code:`log_step_size_reg_target`. Defaults to 0.05 as
recommended in Hoffman and Gelman (2014).
iter_decay_coeff: Coefficient controlling exponent of decay in schedule
weighting stochastic updates to smoothed log step size estimate. Should
@@ -301,15 +301,15 @@ Source code for mici.adapters
> 0 has the effect of stabilising early iterations. Defaults to the
value of 10 as recommended in Hoffman and Gelman (2014).
max_init_step_size_iters: Maximum number of iterations to use in initial
- search for a reasonable step size with an `AdaptationError` exception
- raised if a suitable step size is not found within this many iterations.
+ search for a reasonable step size with an
+ :py:exc:`mici.errors.AdaptationError` exception raised if a suitable
+ step size is not found within this many iterations.
log_step_size_reducer: Reduction to apply to final per-chain step sizes
estimates to produce overall integrator step size for main chain stages.
The specified function should accept a sequence of logarithms of
estimated step sizes and output a non-negative step size to use. If
- `None`, the default, a function which computes the arithmetic mean of
- the step sizes is used.
-
+ :code:`None`, the default, a function which computes the arithmetic mean
+ of the per-chain step sizes is used.
"""
self.adapt_stat_target = adapt_stat_target
self.adapt_stat_func = (
@@ -450,27 +450,27 @@ Source code for mici.adapters
[docs]class OnlineVarianceMetricAdapter(Adapter):
"""Diagonal metric adapter using online variance estimates.
- Uses Welford's algorithm [1] to stably compute an online estimate of the sample
- variances of the chain state position components during sampling. If online
+ Uses Welford's algorithm (Welford, 1962) to stably compute an online estimate of the
+ sample variances of the chain state position components during sampling. If online
estimates are available from multiple independent chains, the final variance
estimate is calculated from the per-chain statistics using the parallel / batched
- incremental variance algorithm described by Chan et al. [2]. The variance estimates
- are optionally regularized towards a common scalar value, with increasing weight for
- small number of samples, to decrease the effect of noisy estimates for small sample
- sizes, following the approach in Stan [3]. The metric matrix representation is set
- to a diagonal matrix with diagonal elements corresponding to the reciprocal of the
- (regularized) variance estimates.
+ incremental variance algorithm described by Chan et al. (1979). The variance
+ estimates are optionally regularized towards a common scalar value, with increasing
+ weight for small number of samples, to decrease the effect of noisy estimates for
+ small sample sizes, following the approach in Stan (Carpenter et al., 2017). The
+ metric matrix representation is set to a diagonal matrix with diagonal elements
+ corresponding to the reciprocal of the (regularized) variance estimates.
References:
- 1. Welford, B. P., 1962. Note on a method for calculating corrected sums of
+ 1. Welford, B. P. (1962). Note on a method for calculating corrected sums of
squares and products. Technometrics, 4(3), pp. 419–420.
- 2. Chan, T. F., Golub, G. H., LeVeque, R. J., 1979. Updating formulae and a
+ 2. Chan, T. F., Golub, G. H. and LeVeque, R. J. (1979). Updating formulae and a
pairwise algorithm for computing sample variances. Technical Report
STAN-CS-79-773, Department of Computer Science, Stanford University.
3. Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt,
- M., Brubaker, M., Guo, J., Li, P. and Riddell, A., 2017. Stan: A probabilistic
- programming language. Journal of Statistical Software, 76(1).
+ M., Brubaker, M., Guo, J., Li, P. and Riddell, A. (2017). Stan: A
+ probabilistic programming language. Journal of Statistical Software, 76(1).
"""
is_fast = False
@@ -558,7 +558,7 @@ Source code for mici.adapters
mean_est /= n_iter
var_est += adapt_state["sum_diff_sq"]
var_est += (
- mean_diff ** 2 * (adapt_state["iter"] * n_iter_prev) / n_iter
+ mean_diff**2 * (adapt_state["iter"] * n_iter_prev) / n_iter
)
if n_iter < 2:
raise AdaptationError(
@@ -576,31 +576,31 @@ Source code for mici.adapters
[docs]class OnlineCovarianceMetricAdapter(Adapter):
"""Dense metric adapter using online covariance estimates.
- Uses Welford's algorithm [1] to stably compute an online estimate of the sample
- covariance matrix of the chain state position components during sampling. If online
- estimates are available from multiple independent chains, the final covariance
- matrix estimate is calculated from the per-chain statistics using a covariance
- variant due to Schubert and Gertz [2] of the parallel / batched incremental variance
- algorithm described by Chan et al. [3]. The covariance matrix estimates are
- optionally regularized towards a scaled identity matrix, with increasing weight for
- small number of samples, to decrease the effect of noisy estimates for small sample
- sizes, following the approach in Stan [4]. The metric matrix representation is set
- to a dense positive definite matrix corresponding to the inverse of the
- (regularized) covariance matrix estimate.
+ Uses Welford's algorithm (Welford, 1962) to stably compute an online estimate of the
+ sample covariance matrix of the chain state position components during sampling. If
+ online estimates are available from multiple independent chains, the final
+ covariance matrix estimate is calculated from the per-chain statistics using a
+ covariance variant due to Schubert and Gertz (2018) of the parallel / batched
+ incremental variance algorithm described by Chan et al. (1979). The covariance
+ matrix estimates are optionally regularized towards a scaled identity matrix, with
+ increasing weight for small number of samples, to decrease the effect of noisy
+ estimates for small sample sizes, following the approach in Stan (Carpenter et al.,
+ 2017). The metric matrix representation is set to a dense positive definite matrix
+ corresponding to the inverse of the (regularized) covariance matrix estimate.
References:
- 1. Welford, B. P., 1962. Note on a method for calculating corrected sums of
+ 1. Welford, B. P. (1962). Note on a method for calculating corrected sums of
squares and products. Technometrics, 4(3), pp. 419–420.
- 2. Schubert, E. and Gertz, M., 2018. Numerically stable parallel computation of
+ 2. Schubert, E. and Gertz, M. (2018). Numerically stable parallel computation of
(co-)variance. ACM. p. 10. doi:10.1145/3221269.3223036.
- 3. Chan, T. F., Golub, G. H., LeVeque, R. J., 1979. Updating formulae and a
+ 3. Chan, T. F., Golub, G. H. and LeVeque, R. J. (1979). Updating formulae and a
pairwise algorithm for computing sample variances. Technical Report
STAN-CS-79-773, Department of Computer Science, Stanford University.
4. Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt,
- M., Brubaker, M., Guo, J., Li, P. and Riddell, A., 2017. Stan: A probabilistic
- programming language. Journal of Statistical Software, 76(1).
+ M., Brubaker, M., Guo, J., Li, P. and Riddell, A. (2017). Stan: A
+ probabilistic programming language. Journal of Statistical Software, 76(1).
"""
is_fast = False
diff --git a/_modules/mici/autodiff.html b/_modules/mici/autodiff.html
index 381c529..7b04cf9 100644
--- a/_modules/mici/autodiff.html
+++ b/_modules/mici/autodiff.html
@@ -75,6 +75,7 @@ Source code for mici.autodiff
from __future__ import annotations
from typing import TYPE_CHECKING
+
import mici.autograd_wrapper as autograd_wrapper
if TYPE_CHECKING:
diff --git a/_modules/mici/autograd_wrapper.html b/_modules/mici/autograd_wrapper.html
index 436d757..e15a3e7 100644
--- a/_modules/mici/autograd_wrapper.html
+++ b/_modules/mici/autograd_wrapper.html
@@ -79,24 +79,19 @@ Source code for mici.autograd_wrapper
AUTOGRAD_AVAILABLE = True
try:
- from autograd.wrap_util import unary_to_nary
+ import autograd.numpy as np
from autograd.builtins import tuple as atuple
from autograd.core import make_vjp
from autograd.extend import vspace
- import autograd.numpy as np
+ from autograd.wrap_util import unary_to_nary
except ImportError:
AUTOGRAD_AVAILABLE = False
if TYPE_CHECKING:
from typing import Callable
- from mici.types import (
- ScalarLike,
- ArrayLike,
- ScalarFunction,
- ArrayFunction,
- MatrixHessianProduct,
- MatrixTressianProduct,
- )
+
+ from mici.types import (ArrayFunction, ArrayLike, MatrixHessianProduct,
+ MatrixTressianProduct, ScalarFunction, ScalarLike)
def _wrapped_unary_to_nary(func: Callable) -> Callable:
diff --git a/_modules/mici/integrators.html b/_modules/mici/integrators.html
index 1448c53..b12556e 100644
--- a/_modules/mici/integrators.html
+++ b/_modules/mici/integrators.html
@@ -76,21 +76,21 @@ Source code for mici.integrators
from abc import ABC, abstractmethod
from typing import TYPE_CHECKING
+
import numpy as np
from numpy.typing import ArrayLike
-from mici.errors import NonReversibleStepError, AdaptationError
-from mici.solvers import (
- maximum_norm,
- solve_fixed_point_direct,
- solve_projection_onto_manifold_quasi_newton,
- FixedPointSolver,
- ProjectionSolver,
-)
+
+from mici.errors import AdaptationError, NonReversibleStepError
+from mici.solvers import (FixedPointSolver, ProjectionSolver, maximum_norm,
+ solve_fixed_point_direct,
+ solve_projection_onto_manifold_newton)
if TYPE_CHECKING:
from typing import Any, Callable, Optional, Sequence
+
from mici.states import ChainState
- from mici.systems import System, TractableFlowSystem
+ from mici.systems import (ConstrainedTractableFlowSystem, System,
+ TractableFlowSystem)
from mici.types import NormFunction
@@ -363,7 +363,7 @@ Source code for mici.integrators
step size adapter will be used to set the step size before calling the
:py:meth:`step` method.
"""
- a_0 = (3 - 3 ** 0.5) / 6
+ a_0 = (3 - 3**0.5) / 6
super().__init__(system, (a_0,), step_size, True)
@@ -463,17 +463,27 @@ Source code for mici.integrators
C(t)(q, p) = (q + t \nabla_2 h_2(q, p), p), \\
C^*(t)(q, p) = \lbrace (q', p) : q' = q + t \nabla_2 h_2(q', p) \rbrace.
- Fixed-point iterations are used to solve the non-linear systems of equations in the
- implicit component updates (:math:`B` and :math:`C^*`).
-
The resulting implicit integrator is a symmetric second-order method corresponding
to a symplectic partitioned Runge-Kutta method. See Section 6.3.2 in Leimkuhler and
Reich (2004) for more details.
+ Fixed-point iterations are used to solve the non-linear systems of equations in the
+ implicit component updates (:math:`B` and :math:`C^*`). As the iterative solves may
+ fail to converge, or may converge to one of multiple solutions, following the
+ approach proposed by Zappa, Holmes-Cerfon and Goodman (2018), an explicit
+ *reversibility check* is performed to ensure the overall integrator step is
+ time-reversible. If the reversibility check fails or the iterative solver fails to
+ converge an appropriate error is raised
+ (:py:exc:`mici.errors.NonReversibleStepError` and
+ :py:exc:`mici.errors.ConvergenceError` respectively).
+
References:
1. Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
Cambridge University Press.
+ 2. Zappa, E., Holmes‐Cerfon, M., & Goodman, J. (2018). Monte Carlo on manifolds:
+ sampling densities and integrating functions. Communications on Pure and
+ Applied Mathematics, 71(12), 2609-2647.
"""
def __init__(
@@ -600,17 +610,26 @@ Source code for mici.integrators
\rbrace, \\
A^*(t)(q, p) = (q + t \nabla_2 h(q, p), p - t \nabla_1 h(q, p)).
- A fixed-point iteration is used to solve the non-linear system of equations in the
- implicit Euler step :math:`A`.
-
The resulting implicit integrator is a second-order method corresponding to a
symplectic one-stage Runge-Kutta method. See Sections 4.1 and 6.3.1 in Leimkuhler
and Reich (2004) for more details.
+ A fixed-point iteration is used to solve the non-linear system of equations in the
+ implicit Euler step :math:`A`. As the iterative solve may fail to converge, or may
+ converge to one of multiple solutions, following the approach proposed by
+ Zappa, Holmes-Cerfon and Goodman (2018), an explicit *reversibility check* is
+ performed to ensure the overall integrator step is time-reversible. If the
+ reversibility check fails or the iterative solver fails to converge an appropriate
+ error is raised (:py:exc:`mici.errors.NonReversibleStepError` and
+ :py:exc:`mici.errors.ConvergenceError` respectively).
+
References:
1. Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
Cambridge University Press.
+ 2. Zappa, E., Holmes‐Cerfon, M., & Goodman, J. (2018). Monte Carlo on manifolds:
+ sampling densities and integrating functions. Communications on Pure and
+ Applied Mathematics, 71(12), 2609-2647.
"""
def __init__(
@@ -712,7 +731,7 @@ Source code for mici.integrators
.. math::
- h(q, p) = h_1(q) + h_2(q, p)
+ h(q, p) = h_1(q) + h_2(q, p),
where :math:`q` and :math:`p` are the position and momentum variables respectively,
and :math:`h_1` and :math:`h_2` Hamiltonian component functions for which the exact
@@ -751,7 +770,7 @@ Source code for mici.integrators
.. math::
- \partial c(q) \nabla_2 h(q, p) = \nabla_2 h_2(q, p) = 0.
+ \partial c(q) \nabla_2 h(q, p) = \partial c(q) \nabla_2 h_2(q, p) = 0.
The set of momentum variables satisfying the above for given position variables is
termed the cotangent space of the manifold (at a position), and the set of
@@ -811,7 +830,8 @@ Source code for mici.integrators
.. math::
- c((\Phi_2(t) \circ \Pi(\lambda)(q, p))_1) = 0
+ c((\Phi_2(t) \circ \Pi(\lambda)(q, p))_1)
+ = c((\Phi_2(t)(q, p + \partial c(q)^T \lambda))_1) = 0,
i.e. solving for the values of the Lagrange multipliers such that the position
component of the output of :math:`\Phi_2(t) \circ \Pi(\lambda)` is on the manifold,
@@ -821,6 +841,8 @@ Source code for mici.integrators
state pair, with this projection step as noted above typically having an analytic
solution.
+ For more details see Reich (1996) and section 7.5.1 in Leimkuhler and Reich (2004).
+
The overall second-order integrator is then defined as the symmetric composition
.. math::
@@ -828,12 +850,30 @@ Source code for mici.integrators
\Psi(t) = A(t / 2) \circ B(t / N)^N \circ A(t / 2),
where :math:`N` is a positive integer corresponding to the number of 'inner'
- :math:`h_2` flow steps. This integrator exactly preserves the constraints at all
- steps, such that if an initial position momentum pair :math:`(q, p)` are in the
- cotangent bundle, the corresponding pair after calling the :py:meth:`step` method of
- the integrator will also be in the cotangent bundle.
-
- For more details see Reich (1996) and section 7.5.1 in Leimkuhler and Reich (2004).
+ :math:`h_2` flow steps, following the 'geodesic integrator' formulation proposed by
+ Leimkuhler and Matthews (2016). The additional flexibility introduced by having
+ the possibility of :math:`N > 1` is particularly of use when evaluation of
+ :math:`\Phi_1` is significantly more expensive than evaluation of :math:`\Phi_2`; in
+ this case using :math:`N > 1` can allow a larger time step :math:`t` to be used than
+ may be otherwise possible due to the need to ensure the iterative solver used in
+ :math:`B` does not diverge, with a smaller step size :math:`t / N` used for the
+ steps involving the iterative solves with the (cheaper) :math:`\Phi_2` flow map
+ and a larger step size :math:`t` used for the steps involving the (more expensive)
+ :math:`\Phi_1` flow map.
+
+ This integrator exactly preserves the constraints at all steps, such that if an
+ initial position momentum pair :math:`(q, p)` are in the cotangent bundle, the
+ corresponding pair after calling the :py:meth:`step` method of the integrator will
+ also be in the cotangent bundle, *providing the iterative solver converges*.
+
+ As the iterative solves may fail to converge, or may converge to one of multiple
+ solutions, following the approach proposed by Zappa, Holmes-Cerfon and Goodman
+ (2018), an explicit *reversibility check* is performed to ensure the overall
+ integrator step is time-reversible; see also Lelievre, Rousset and Stoltz (2019) for
+ an analysis of this approach specifically in the context of Hamiltonian Monte Carlo.
+ If the reversibility check fails or the iterative solver fails to converge an
+ appropriate error is raised (:py:exc:`mici.errors.NonReversibleStepError` and
+ :py:exc:`mici.errors.ConvergenceError` respectively).
References:
@@ -841,21 +881,31 @@ Source code for mici.integrators
composition methods. SIAM journal on numerical analysis, 33(2), 475-491.
2. Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
Cambridge University Press.
+ 3. Leimkuhler, B., & Matthews, C. (2016). Efficient molecular dynamics using
+ geodesic integration and solvent–solute splitting. Proceedings of the Royal
+ Society A: Mathematical, Physical and Engineering Sciences, 472(2189),
+ 20160138.
+ 4. Zappa, E., Holmes‐Cerfon, M., & Goodman, J. (2018). Monte Carlo on manifolds:
+ sampling densities and integrating functions. Communications on Pure and
+ Applied Mathematics, 71(12), 2609-2647.
+ 5. Lelievre, T., Rousset, M., & Stoltz, G. (2019). Hybrid Monte Carlo methods for
+ sampling probability measures on submanifolds. Numerische Mathematik, 143,
+ 379-421.
"""
def __init__(
self,
- system: System,
+ system: ConstrainedTractableFlowSystem,
step_size: Optional[float] = None,
n_inner_step: int = 1,
reverse_check_tol: float = 2e-8,
reverse_check_norm: NormFunction = maximum_norm,
- projection_solver: ProjectionSolver = solve_projection_onto_manifold_quasi_newton,
+ projection_solver: ProjectionSolver = solve_projection_onto_manifold_newton,
projection_solver_kwargs: Optional[dict[str, Any]] = None,
):
"""
Args:
- system: Hamiltonian system to integrate the dynamics of.
+ system: Hamiltonian system to integrate the constrained dynamics of.
step_size: Integrator time step. If set to :code:`None` it is assumed that a
step size adapter will be used to set the step size before calling the
:py:meth:`step` method.
@@ -894,7 +944,7 @@ Source code for mici.integrators
.. code::
system.constr(
- state.pos
+ state.pos
+ dh2_flow_pos_dmom @ system.jacob_constr(state_prev).T @ λ
) == 0
@@ -902,7 +952,9 @@ Source code for mici.integrators
the derivative of the action of the (linear) :code:`system.h2_flow` map
on the state momentum component with respect to the position component.
This is used to project the state position component back on to the
- manifold after an unconstrained :code:`system.h2_flow` update.
+ manifold after an unconstrained :code:`system.h2_flow` update. If the
+ solver fails to convege a :py:exc:`mici.errors.ConvergenceError`
+ exception is raised.
projection_solver_kwargs: Dictionary of any keyword arguments to
:code:`projection_solver`.
"""
diff --git a/_modules/mici/interop.html b/_modules/mici/interop.html
index 67e70ba..3954cfa 100644
--- a/_modules/mici/interop.html
+++ b/_modules/mici/interop.html
@@ -74,16 +74,20 @@ Source code for mici.interop
from __future__ import annotations
+import importlib
import os
-import mici
-import numpy as np
from typing import TYPE_CHECKING
+import numpy as np
+
+import mici
+
if TYPE_CHECKING:
from typing import Literal, Optional, Union
- from numpy.typing import ArrayLike
+
import arviz
- import pymc3
+ import pymc
+ from numpy.typing import ArrayLike
[docs]def convert_to_inference_data(
@@ -92,10 +96,10 @@ Source code for mici.interop
energy_key: Optional[str] = "energy",
lp_key: Optional[str] = "lp",
) -> arviz.InferenceData:
- """Convert Mici :code:`sample_chains` output to :py:class:`arviz.InferenceData` object.
+ """Convert Mici :code:`sample_chains` output to :py:class:`arviz.InferenceData`.
Args:
- traces: Traces output from Mici
+ traces: Traces output from Mici
:py:meth:`mici.samplers.MarkovChainMonteCarloMethod.sample_chains` call. A
dictionary of variables traced over sampled chains with the dictionary keys
the variable names and the values a list of arrays, one array per sampled
@@ -133,7 +137,7 @@ Source code for mici.interop
)
-[docs]def sample_pymc3_model(
+[docs]def sample_pymc_model(
draws: int = 1000,
*,
tune: int = 1000,
@@ -145,16 +149,16 @@ Source code for mici.interop
jitter_max_retries: int = 10,
trace: Optional[list] = None,
return_inferencedata: bool = False,
- model: Optional[pymc3.model.Model] = None,
+ model: Optional[pymc.Model] = None,
target_accept: float = 0.8,
max_treedepth: int = 10,
) -> Union[arviz.InferenceData, dict[str, ArrayLike]]:
- """Generate approximate samples from posterior defined by a PyMC3 model.
+ """Generate approximate samples from posterior defined by a PyMC model.
Uses dynamic multinomial HMC algorithm in Mici with adaptive warm-up phase.
- This function replicates the interface of the :py:func:`pymc3.sampling.sample`
- function to allow using as a (partial) drop-in replacement.
+ This function replicates the interface of the :py:func:`pymc.sample` function to
+ allow using as a (partial) drop-in replacement.
Args:
draws: The number of samples to draw.
@@ -192,7 +196,7 @@ Source code for mici.interop
return_inferencedata: Whether to return the traces as an
:py:class:`arviz.InferenceData` (:code:`True`) object or a
:py:class:`dict` (:code:`False`).
- model: PyMC3 model defining posterior distribution to sample from. May be
+ model: PyMC model defining posterior distribution to sample from. May be
:code:`None` if function is called from within model context manager.
target_accept: Target value for the acceptance statistic being controlled during
adaptive warm-up.
@@ -214,16 +218,12 @@ Source code for mici.interop
the :code:`sample_stats` group.
"""
- import pymc3
+ import pymc
- if return_inferencedata:
-
- try:
- import arviz
- except ImportError:
- raise ValueError("Cannot return InferenceData as ArviZ is not installed")
+ if return_inferencedata and importlib.util.find_spec("arviz") is None:
+ raise ValueError("Cannot return InferenceData as ArviZ is not installed")
- model = pymc3.modelcontext(model)
+ model = pymc.modelcontext(model)
if cores is None:
cores = min(4, os.cpu_count() // 2) # assume 2 threads per CPU core
@@ -248,6 +248,9 @@ Source code for mici.interop
'init must be "auto", "jitter+adapt_diag", "adapt_diag" or "adapt_full"'
)
+ initial_point = model.initial_point()
+ raveled_initial_point = pymc.blocking.DictToArrayBijection.map(initial_point)
+
val_and_grad_log_dens = model.logp_dlogp_function()
val_and_grad_log_dens.set_extra_values({})
@@ -260,14 +263,18 @@ Source code for mici.interop
return -val
def trace_func(state):
- var_dict = val_and_grad_log_dens.array_to_dict(state.pos)
+ raveled_vars = pymc.blocking.RaveledVars(
+ state.pos, raveled_initial_point.point_map_info
+ )
+ var_dict = pymc.blocking.DictToArrayBijection.rmap(raveled_vars)
trace_dict = {}
for rv in trace:
if rv.name in var_dict:
trace_dict[rv.name] = var_dict[rv.name]
else:
- trace_dict[rv.name] = rv.transformation.backward(
- var_dict[rv.transformed.name]
+ transform = model.rvs_to_transforms[rv]
+ trace_dict[rv.name] = transform.backward(
+ var_dict[f"{rv.name}_{transform.name}__"], *rv.owner.inputs
).eval()
trace_dict["lp"] = -system.neg_log_dens(state)
trace_dict["energy"] = system.h(state)
@@ -280,7 +287,7 @@ Source code for mici.interop
integrator = mici.integrators.LeapfrogIntegrator(system)
rng = np.random.default_rng(random_seed)
sampler = mici.samplers.DynamicMultinomialHMC(
- system, integrator, rng, max_treedepth=max_treedepth
+ system, integrator, rng, max_tree_depth=max_treedepth
)
step_size_adapter = mici.adapters.DualAveragingStepSizeAdapter(target_accept)
@@ -292,7 +299,7 @@ Source code for mici.interop
adapters = [step_size_adapter, metric_adapter]
if jitter_init:
- mean = val_and_grad_log_dens.dict_to_array(model.test_point)
+ mean = raveled_initial_point.data.copy()
init_states = []
for c in range(chains):
for t in range(jitter_max_retries):
@@ -301,9 +308,7 @@ Source code for mici.interop
break
init_states.append(pos)
else:
- init_states = [
- val_and_grad_log_dens.dict_to_array(model.test_point) for c in range(chains)
- ]
+ init_states = [raveled_initial_point.data.copy() for c in range(chains)]
_, traces, stats = sampler.sample_chains(
n_warm_up_iter=tune,
@@ -410,12 +415,8 @@ Source code for mici.interop
import stan
- if return_inferencedata:
-
- try:
- import arviz
- except ImportError:
- raise ValueError("Cannot return InferenceData as ArviZ is not installed")
+ if return_inferencedata and importlib.util.find_spec("arviz") is None:
+ raise ValueError("Cannot return InferenceData as ArviZ is not installed")
model = stan.build(model_code, data=data)
@@ -488,7 +489,7 @@ Source code for mici.interop
stager=stager,
trace_funcs=[trace_func],
monitor_stats=["accept_stat", "n_step", "diverging"],
- trace_warmup=save_warmup,
+ trace_warm_up=save_warmup,
)
if return_inferencedata:
diff --git a/_modules/mici/matrices.html b/_modules/mici/matrices.html
index a7aca08..c88b3f1 100644
--- a/_modules/mici/matrices.html
+++ b/_modules/mici/matrices.html
@@ -77,15 +77,19 @@ Source code for mici.matrices
import abc
import numbers
from typing import TYPE_CHECKING
+
import numpy as np
-from mici.errors import LinAlgError
import numpy.linalg as nla
import scipy.linalg as sla
+
+from mici.errors import LinAlgError
from mici.utils import hash_array
if TYPE_CHECKING:
from typing import Iterable, Literal, Optional, Tuple, Union
+
from numpy.typing import NDArray
+
from mici.types import MatrixLike, ScalarLike
@@ -720,7 +724,7 @@ Source code for mici.matrices
return self.shape[0] / self._scalar
[docs] def grad_quadratic_form_inv(self, vector: NDArray) -> float:
- return -np.sum(vector ** 2) / self._scalar ** 2
+ return -np.sum(vector**2) / self._scalar**2
def __str__(self) -> str:
return f"(shape={self.shape}, scalar={self._scalar})"
@@ -753,7 +757,7 @@ Source code for mici.matrices
return PositiveScaledIdentityMatrix(1 / self._scalar, self.shape[0])
def _construct_sqrt(self) -> PositiveScaledIdentityMatrix:
- return PositiveScaledIdentityMatrix(self._scalar ** 0.5, self.shape[0])
+ return PositiveScaledIdentityMatrix(self._scalar**0.5, self.shape[0])
[docs]class DiagonalMatrix(SymmetricMatrix, DifferentiableMatrix, ImplicitArrayMatrix):
@@ -838,7 +842,7 @@ Source code for mici.matrices
return PositiveDiagonalMatrix(1.0 / self.diagonal)
def _construct_sqrt(self) -> PositiveDiagonalMatrix:
- return PositiveDiagonalMatrix(self.diagonal ** 0.5)
+ return PositiveDiagonalMatrix(self.diagonal**0.5)
def _make_array_triangular(array: NDArray, lower: bool) -> NDArray:
@@ -1114,7 +1118,7 @@ Source code for mici.matrices
def _scalar_multiply(self, scalar: ScalarLike) -> TriangularFactoredDefiniteMatrix:
if scalar > 0:
return TriangularFactoredPositiveDefiniteMatrix(
- factor=scalar ** 0.5 * self.factor
+ factor=scalar**0.5 * self.factor
)
else:
return super()._scalar_multiply(scalar)
@@ -1614,7 +1618,7 @@ Source code for mici.matrices
return EigendecomposedPositiveDefiniteMatrix(self.eigvec, 1 / self.eigval)
def _construct_sqrt(self) -> EigendecomposedPositiveDefiniteMatrix:
- return EigendecomposedPositiveDefiniteMatrix(self.eigvec, self.eigval ** 0.5)
+ return EigendecomposedPositiveDefiniteMatrix(self.eigvec, self.eigval**0.5)
[docs]class SoftAbsRegularizedPositiveDefiniteMatrix(
diff --git a/_modules/mici/progressbars.html b/_modules/mici/progressbars.html
index 6f5198a..158a7f6 100644
--- a/_modules/mici/progressbars.html
+++ b/_modules/mici/progressbars.html
@@ -297,7 +297,7 @@ Source code for mici.progressbars
@property
def description(self) -> str:
- """"Description of task being tracked."""
+ """Description of task being tracked."""
return self._description
@property
diff --git a/_modules/mici/samplers.html b/_modules/mici/samplers.html
index e04f7a1..ca3b68a 100644
--- a/_modules/mici/samplers.html
+++ b/_modules/mici/samplers.html
@@ -74,62 +74,49 @@ Source code for mici.samplers
from __future__ import annotations
+import logging
import os
import queue
+import signal
+import tempfile
from contextlib import ExitStack, contextmanager, nullcontext
from pathlib import Path
from pickle import PicklingError
-import logging
-import tempfile
-import signal
-from warnings import warn
from typing import TYPE_CHECKING, NamedTuple
+from warnings import warn
+
import numpy as np
from numpy.random import default_rng
-from mici.transitions import (
- IndependentMomentumTransition,
- MetropolisRandomIntegrationTransition,
- MetropolisStaticIntegrationTransition,
- MultinomialDynamicIntegrationTransition,
- SliceDynamicIntegrationTransition,
- euclidean_no_u_turn_criterion,
- riemannian_no_u_turn_criterion,
-)
-from mici.states import ChainState
-from mici.progressbars import (
- SequenceProgressBar,
- LabelledSequenceProgressBar,
- DummyProgressBar,
- _ProxySequenceProgressBar,
-)
-from mici.errors import AdaptationError
+
from mici.adapters import DualAveragingStepSizeAdapter
+from mici.errors import AdaptationError
+from mici.progressbars import (DummyProgressBar, LabelledSequenceProgressBar,
+ SequenceProgressBar, _ProxySequenceProgressBar)
from mici.stagers import WarmUpStager, WindowedWarmUpStager
+from mici.states import ChainState
+from mici.transitions import (IndependentMomentumTransition,
+ MetropolisRandomIntegrationTransition,
+ MetropolisStaticIntegrationTransition,
+ MultinomialDynamicIntegrationTransition,
+ SliceDynamicIntegrationTransition,
+ euclidean_no_u_turn_criterion,
+ riemannian_no_u_turn_criterion)
if TYPE_CHECKING:
- from typing import (
- Container,
- Generator,
- Iterable,
- Optional,
- Sequence,
- Union,
- )
+ from typing import (Container, Generator, Iterable, Optional, Sequence,
+ Union)
+
from numpy.typing import ArrayLike, DTypeLike, NDArray
+
from mici.adapters import Adapter
from mici.integrators import Integrator
from mici.progressbars import ProgressBar
from mici.stagers import Stager
from mici.systems import System
- from mici.transitions import IntegrationTransition, MomentumTransition, Transition
- from mici.types import (
- AdapterState,
- ChainIterator,
- ScalarLike,
- PyTree,
- TraceFunction,
- TerminationCriterion,
- )
+ from mici.transitions import (IntegrationTransition, MomentumTransition,
+ Transition)
+ from mici.types import (AdapterState, ChainIterator, PyTree, ScalarLike,
+ TerminationCriterion, TraceFunction)
# Preferentially import from multiprocess library if available as able to
# serialize much wider range of types including autograd functions
@@ -217,7 +204,10 @@ Source code for mici.samplers
def _open_new_memmap(
- file_path: str, shape: tuple[int, ...], default_val: ScalarLike, dtype: DTypeLike,
+ file_path: str,
+ shape: tuple[int, ...],
+ default_val: ScalarLike,
+ dtype: DTypeLike,
) -> np.memmap:
"""Open a new memory-mapped array object and fill with a default-value.
@@ -370,7 +360,15 @@ Source code for mici.samplers
]
else:
traces[key] = list(
- np.full((n_chain, n_iter,) + val.shape, init, val.dtype)
+ np.full(
+ (
+ n_chain,
+ n_iter,
+ )
+ + val.shape,
+ init,
+ val.dtype,
+ )
)
return traces
@@ -1033,7 +1031,11 @@ Source code for mici.samplers
)
)
stats = _init_stats(
- self.transitions, n_chain, n_trace_iter, use_memmap, memmap_path,
+ self.transitions,
+ n_chain,
+ n_trace_iter,
+ use_memmap,
+ memmap_path,
)
per_chain_rngs = _get_per_chain_rngs(self.rng, n_chain)
per_chain_traces = (
@@ -1123,7 +1125,7 @@ Source code for mici.samplers
and main sampling stages otherwise. The key for each value is the
corresponding key in the dictionary returned by the trace function which
computed the traced value.
- statistics: Dictionary of chain transition statistic dictionaries. Values in
+ statistics: Dictionary of chain transition statistic dictionaries. Values in
dictionary are lists of arrays of chain statistic values with each array in
the list corresponding to a single chain and the leading dimension of each
array corresponding to the iteration (draw) index, within the main
diff --git a/_modules/mici/solvers.html b/_modules/mici/solvers.html
index 242a4bc..d48b584 100644
--- a/_modules/mici/solvers.html
+++ b/_modules/mici/solvers.html
@@ -74,19 +74,22 @@ Source code for mici.solvers
from __future__ import annotations
-from typing import Protocol, TYPE_CHECKING
-from mici.errors import ConvergenceError, LinAlgError
+from typing import TYPE_CHECKING, Protocol
+
import numpy as np
+from mici.errors import ConvergenceError, LinAlgError
+
if TYPE_CHECKING:
from mici.states import ChainState
- from mici.systems import System
- from mici.types import ScalarFunction, ArrayFunction, ArrayLike
+ from mici.systems import (ConstrainedEuclideanMetricSystem,
+ ConstrainedTractableFlowSystem)
+ from mici.types import ArrayFunction, ArrayLike, ScalarFunction
[docs]def euclidean_norm(vct):
"""Calculate the Euclidean (L-2) norm of a vector."""
- return (vct ** 2).sum() ** 0.5
+ return (vct**2).sum() ** 0.5
[docs]def maximum_norm(vct):
@@ -95,7 +98,7 @@ Source code for mici.solvers
[docs]class FixedPointSolver(Protocol):
- """Solver for fixed point equation, returning `x` such that `func(x) = x`."""
+ """Solver for fixed point equation :code:`func(x) = x`."""
[docs] def __call__(self, func: ArrayFunction, x0: ArrayLike, **kwargs) -> ArrayLike:
"""Solve fixed point equation.
@@ -117,24 +120,26 @@ Source code for mici.solvers
max_iters: int = 100,
norm: ScalarFunction = maximum_norm,
) -> ArrayLike:
- """Solve fixed point equation `func(x) = x` using direct iteration.
+ """Solve fixed point equation :code:`func(x) = x` using direct iteration.
Args:
func: Function to find fixed point of.
x0: Initial state (function argument).
convergence_tol: Convergence tolerance - solver successfully terminates when
- `norm(func(x) - x) < convergence_tol`.
+ :code:`norm(func(x) - x) < convergence_tol`.
divergence_tol: Divergence tolerance - solver aborts if
- `norm(func(x) - x) > divergence_tol` on any iteration.
+ :code:`norm(func(x) - x) > divergence_tol` on any iteration.
max_iters: Maximum number of iterations before raising exception.
norm: Norm to use to assess convergence.
Returns:
- Solution to fixed point equation with `norm(func(x) - x) < convergence_tol`.
+ Solution to fixed point equation with
+ :code:`norm(func(x) - x) < convergence_tol`.
Raises:
- mici.errors.ConvergenceError: If solver does not converge within `max_iters`
- iterations, diverges or encounters a `ValueError` during the iteration.
+ mici.errors.ConvergenceError: If solver does not converge within
+ :code:`max_iters` iterations, diverges or encounters a :py:exc:`ValueError`
+ during the iteration.
"""
for i in range(max_iters):
try:
@@ -166,30 +171,30 @@ Source code for mici.solvers
max_iters: int = 100,
norm: ScalarFunction = maximum_norm,
) -> ArrayLike:
- """Solve fixed point equation `func(x) = x` using Steffensen's method.
-
- Steffennsen's method [1] achieves quadratic convergence but at the cost of two
- function evaluations per iteration so for functions where convergence is achieved in
- a small number of iterations, direct iteration may be cheaper.
+ """Solve fixed point equation :code:`func(x) = x` using Steffensen's method.
- [1] : https://en.wikipedia.org/wiki/Steffensen%27s_method
+ Steffennsen's method achieves quadratic convergence but at the cost of two function
+ evaluations per iteration so for functions where convergence is achieved in a small
+ number of iterations, direct iteration may be cheaper.
Args:
func: Function to find fixed point of.
x0: Initial state (function argument).
convergence_tol: Convergence tolerance - solver successfully terminates
- when `norm(func(x) - x) < convergence_tol`.
+ when :code:`norm(func(x) - x) < convergence_tol`.
divergence_tol: Divergence tolerance - solver aborts if
- `norm(func(x) - x) > divergence_tol` on any iteration.
+ :code:`norm(func(x) - x) > divergence_tol` on any iteration.
max_iters: Maximum number of iterations before raising exception.
norm: Norm to use to assess convergence.
Returns:
- Solution to fixed point equation with `norm(func(x) - x) < convergence_tol`.
+ Solution to fixed point equation with
+ :code:`norm(func(x) - x) < convergence_tol`.
Raises:
- mici.errors.ConvergenceError: If solver does not converge within `max_iters`
- iterations, diverges or encounters a `ValueError` during the iteration.
+ mici.errors.ConvergenceError: If solver does not converge within
+ :code:`max_iters` iterations, diverges or encounters a :py:exc:`ValueError`
+ during the iteration.
"""
for i in range(max_iters):
try:
@@ -220,18 +225,19 @@ Source code for mici.solvers
[docs]class ProjectionSolver(Protocol):
- """Solver for projection on to manifold step in constrained integrator.
+ r"""Solver for projection on to manifold step in constrained integrator.
Solves an equation of the form
- .. code::
+ .. math::
- r(λ) = c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) = 0
+ r(\lambda) = c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda)) = 0,
- for the vector of Lagrange multipliers `λ` to project a point on to the manifold
- defined by the zero level set of a constraint function `c` (`system.constr`), with
- `Φ₂,₁` the position component of a flow map for the `h₂` Hamiltonian component for
- the system (i.e. `system.h2_flow`).
+ for the vector of Lagrange multipliers :math:`\lambda` to project a point on to the
+ manifold defined by the zero level set of a constraint function :math:`c`, with
+ :math:`\Phi_{2,1}` the flow map for the :math:`h_2` Hamiltonian component for the
+ system restricted to the position component output. The map :math:`\Phi_{2,1}` is
+ assumed to be linear in its second (momentum) argument.
"""
[docs] def __call__(
@@ -239,7 +245,7 @@ Source code for mici.solvers
state: ChainState,
state_prev: ChainState,
time_step: float,
- system: System,
+ system: ConstrainedTractableFlowSystem,
**kwargs,
) -> ChainState:
"""Solve for projection on to manifold step.
@@ -248,7 +254,7 @@ Source code for mici.solvers
state: Current chain state after unconstrained step.
state_prev: Previous chain state on manifold.
time_step: Integrator time step for unconstrained step.
- system: Hamiltonian system dynamics are being simulated for.
+ system: Hamiltonian system constrained dynamics are being simulated for.
Returns:
Chain state after projection on to manifold.
@@ -260,93 +266,120 @@ Source code for mici.solvers
state: ChainState,
state_prev: ChainState,
time_step: float,
- system: System,
+ system: ConstrainedEuclideanMetricSystem,
constraint_tol: float = 1e-9,
position_tol: float = 1e-8,
divergence_tol: float = 1e10,
max_iters: int = 50,
norm: ScalarFunction = maximum_norm,
) -> ChainState:
- """Solve constraint equation using symmetric quasi-Newton method.
+ r"""Solve constraint equation using symmetric quasi-Newton method.
- Only requires re-evaluating the constraint function `system.constr` within the
+ Only requires re-evaluating the constraint function :code:`system.constr` within the
solver loop and no recomputation of matrix decompositions on each iteration.
Solves an equation of the form
- .. code::
+ .. math::
+
+ r(\lambda) = c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda)) = 0,
- r(λ) = c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) = 0
+ for the vector of Lagrange multipliers :math:`\lambda` to project a point on to the
+ manifold defined by the zero level set of a constraint function :math:`c`, with
+ :math:`\Phi_{2,1}` the flow map for the :math:`h_2` Hamiltonian component for the
+ system restricted to the position component output.
- for the vector of Lagrange multipliers `λ` to project a point on to the manifold
- defined by the zero level set of a constraint function `c` (`system.constr`), with
- `Φ₂,₁` the position component of a flow map for the `h₂` Hamiltonian component for
- the system (i.e. `system.h2_flow`).
+ The Jacobian of the residual function :math:`r` is
- The Jacobian of the residual function `r` is
+ .. math::
- .. code::
+ \partial r(\lambda) =
+ \partial c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda))
+ \partial_2 (\Phi_{2,1}(t))(q, p + \partial c(q)^T \lambda)
+ \partial c(q)^T
- ∂r(λ) = ∂c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) ∂₂Φ₂,₁ ∂c(q)ᵀ
+ where :math:`\partial_2 (\Phi_{2,1}(t))` is the Jacobian of the (position
+ restricted) flow-map :math:`\Phi_{2,1}` with respect to its second (momentum)
+ argument. We assume here that :math:`\Phi_{2,1}(t)` is linear in its second
+ (momentum) argument, such that :math:`\partial_2 (\Phi_{2,1}(t))(q, p + \partial
+ c(q)^T \lambda)` is constant with respect to :math:`\lambda`, and henceforth use the
+ shorthand :math:`\partial_2 (\Phi_{2,1}(t))` to refer to this matrix.
- where `∂₂Φ₂,₁` is the Jacobian of the (linear) flow-map `Φ₂,₁` with respect to the
- second (momentum argument), such that the full Newton update is
+ The full Newton update is
- .. code::
+ .. math::
- λ_(α) = λ - ∂r(λ)⁻¹ r(λ)
+ \lambda' = \lambda - \partial r(\lambda)^{-1} r(\lambda)
- which requires evaluating `∂c` (`system.jacob_constr`) on each iteration and solving
- a linear system in the residual Jacobian `∂r(λ)`.
+ which requires evaluating :math:`\partial c` on each iteration and solving a linear
+ system in the residual Jacobian :math:`\partial r(\lambda)`.
The symmetric quasi-Newton iteration instead uses the approximation
- .. code::
+ .. math::
- ∂c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) ∂₂Φ₂,₁ ∂c(q)ᵀ ≈ ∂c(q) ∂₂Φ₂,₁ ∂c(q)ᵀ
+ \partial c(\Phi_{2,1}(t)(q, p + \partial c(q)^T\lambda))
+ \partial_2 (\Phi_{2,1}(t))
+ \partial c(q)^T
+ \approx
+ \partial c(q)
+ \partial_2 (\Phi_{2,1}(t))
+ \partial c(q)^T
with the corresponding update
- .. code::
+ .. math::
- λ = λ - (∂c(q) ∂₂Φ₂,₁ ∂c(q)ᵀ)⁻¹ r(λ)
+ \lambda' =
+ \lambda
+ - (\partial c(q) \partial_2 (\Phi_{2,1}(t)) \partial c(q)^T)^{-1} r(\lambda)
- allowing a previously computed decomposition of the matrix `∂c(q) ∂₂Φ₂,₁ ∂c(q)ᵀ` to
- be used to solve the linear system in each iteration with no requirement to evaluate
- `∂c` (`system.jacob_constr`) on each iteration.
+ allowing a previously computed decomposition of the matrix
+
+ .. math::
+
+ \partial c(q) \partial_2 (\Phi_{2,1}(t)) \partial c(q)^T,
+
+ to be used to solve the linear system in each iteration with no requirement to
+ evaluate :math:`\partial c` (:code:`system.jacob_constr`) on each iteration.
Args:
- state: Post `h2_flow` update state to project.
- state_prev: Previous state in co-tangent bundle before `h2_flow` update which
- defines the co-tangent space to perform projection in.
- time_step: Integrator time step used in `h2_flow` update.
- system: Hamiltonian system defining `h2_flow` and `constr` functions used to
- define constraint equation to solve.
+ state: Post :code:`h2_flow` update state to project.
+ state_prev: Previous state in co-tangent bundle before :code:`h2_flow` update
+ which defines the co-tangent space to perform projection in.
+ time_step: Integrator time step used in :code:`h2_flow` update.
+ system: Hamiltonian system defining :code:`h2_flow` and :code:`constr` functions
+ used to define constraint equation to solve.
constraint_tol: Convergence tolerance in constraint space. Iteration will
- continue until `norm(constr(pos)) < constraint_tol` where `pos` is the
- position at the current iteration.
+ continue until :code:`norm(constr(pos)) < constraint_tol` where :code:`pos`
+ is the position at the current iteration.
position_tol: Convergence tolerance in position space. Iteration will continue
- until `norm(delt_pos) < position_tol` where `delta_pos` is the change in the
- position in the current iteration.
+ until :code:`norm(delta_pos) < position_tol` where :code:`delta_pos` is the
+ change in the position in the current iteration.
divergence_tol: Divergence tolerance - solver aborts if
- `norm(constr(pos)) > divergence_tol` on any iteration where `pos` is the
- position at the current iteration and raises `mici.errors.ConvergenceError`.
+ :code:`norm(constr(pos)) > divergence_tol` on any iteration where
+ :code:`pos` is the position at the current iteration and raises
+ :py:exc:`mici.errors.ConvergenceError`.
max_iters: Maximum number of iterations to perform before aborting and raising
- `mici.errors.ConvergenceError`.
+ :py:exc:`mici.errors.ConvergenceError`.
norm: Norm to use to test for convergence.
Returns:
Updated state object with position component satisfying constraint equation to
- within `constraint_tol`, i.e. `norm(system.constr(state.pos)) < constraint_tol`.
+ within :code:`constraint_tol`, i.e.
+ :code:`norm(system.constr(state.pos)) < constraint_tol`.
Raises:
- mici.errors.ConvergenceError: If solver does not converge within `max_iters`
- iterations, diverges or encounters a `ValueError` during the iteration.
+ mici.errors.ConvergenceError: If solver does not converge within
+ :code:`max_iters` iterations, diverges or encounters a :py:exc:`ValueError`
+ during the iteration.
"""
mu = np.zeros_like(state.pos)
jacob_constr_prev = system.jacob_constr(state_prev)
# Use absolute value of dt and adjust for sign of dt in mom update below
- dh2_flow_pos_dmom, dh2_flow_mom_dmom = system.dh2_flow_dmom(abs(time_step))
+ dh2_flow_pos_dmom, dh2_flow_mom_dmom = system.dh2_flow_dmom(
+ state_prev, abs(time_step)
+ )
inv_jacob_constr_inner_product = system.jacob_constr_inner_product(
jacob_constr_prev, dh2_flow_pos_dmom
).inv
@@ -382,78 +415,91 @@ Source code for mici.solvers
state: ChainState,
state_prev: ChainState,
time_step: float,
- system: System,
+ system: ConstrainedEuclideanMetricSystem,
constraint_tol: float = 1e-9,
position_tol: float = 1e-8,
divergence_tol: float = 1e10,
max_iters: int = 50,
norm: ScalarFunction = maximum_norm,
) -> ChainState:
- """Solve constraint equation using Newton's method.
+ r"""Solve constraint equation using Newton's method.
- Requires re-evaluating both the constraint function `system.constr` and constraint
- Jacobian `system.jacob_constr` within the solver loop and computation of matrix
- decompositions on each iteration.
+ Requires re-evaluating both the constraint function :code:`system.constr` and
+ constraint Jacobian :code:`system.jacob_constr` within the solver loop and
+ computation of matrix decompositions on each iteration.
Solves an equation of the form
- .. code::
+ .. math::
- r(λ) = c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) = 0
+ r(\lambda) = c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda)) = 0,
- for the vector of Lagrange multipliers `λ` to project a point on to the manifold
- defined by the zero level set of a constraint function `c` (`system.constr`), with
- `Φ₂,₁` the position component of a flow map for the `h₂` Hamiltonian component for
- the system (i.e. `system.h2_flow`).
+ for the vector of Lagrange multipliers :math:`\lambda` to project a point on to the
+ manifold defined by the zero level set of a constraint function :math:`c`, with
+ :math:`\Phi_{2,1}` the flow map for the :math:`h_2` Hamiltonian component for the
+ system restricted to the position component output.
- The Jacobian of the residual function `r` is
+ The Jacobian of the residual function :math:`r` is
- .. code::
+ .. math::
- ∂r(λ) = ∂c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) ∂₂Φ₂,₁ ∂c(q)ᵀ
+ \partial r(\lambda) =
+ \partial c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda))
+ \partial_2 (\Phi_{2,1}(t))(q, p + \partial c(q)^T \lambda)
+ \partial c(q)^T
- where `∂₂Φ₂,₁` is the Jacobian of the (linear) flow-map `Φ₂,₁` with respect to the
- second (momentum argument), such that the Newton update is
+ where :math:`\partial_2 (\Phi_{2,1}(t))` is the Jacobian of the (position
+ restricted) flow-map :math:`\Phi_{2,1}` with respect to its second (momentum)
+ argument. We assume here that :math:`\Phi_{2,1}(t)` is linear in its second
+ (momentum) argument, such that :math:`\partial_2 (\Phi_{2,1}(t))(q, p + \partial
+ c(q)^T \lambda)` is constant with respect to :math:`\lambda`.
- .. code::
+ The full Newton update is
- λ = λ - ∂r(λ)⁻¹ r(λ)
+ .. math::
- which requires evaluating `∂c` (`system.jacob_constr`) on each iteration and solving
- a linear system in the residual Jacobian `∂r(λ)`.
+ \lambda' = \lambda - \partial r(\lambda)^{-1} r(\lambda)
+
+ which requires evaluating :math:`\partial c` on each iteration and solving a linear
+ system in the residual Jacobian :math:`\partial r(\lambda)`.
Args:
- state: Post `h2_flow` update state to project.
- state_prev: Previous state in co-tangent bundle before `h2_flow` update which
- defines the co-tangent space to perform projection in.
- time_step: Integrator time step used in `h2_flow` update.
- system: Hamiltonian system defining `h2_flow` and `constr` functions used to
- define constraint equation to solve.
+ state: Post :code:`h2_flow` update state to project.
+ state_prev: Previous state in co-tangent bundle before :code:`h2_flow` update
+ which defines the co-tangent space to perform projection in.
+ time_step: Integrator time step used in :code:`h2_flow` update.
+ system: Hamiltonian system defining :code:`h2_flow` and :code:`constr` functions
+ used to define constraint equation to solve.
constraint_tol: Convergence tolerance in constraint space. Iteration will
- continue until `norm(constr(pos)) < constraint_tol` where `pos` is the
- position at the current iteration.
+ continue until :code:`norm(constr(pos)) < constraint_tol` where :code:`pos`
+ is the position at the current iteration.
position_tol: Convergence tolerance in position space. Iteration will continue
- until `norm(delt_pos) < position_tol` where `delta_pos` is the change in the
- position in the current iteration.
+ until :code:`norm(delta_pos) < position_tol` where :code:`delta_pos` is the
+ change in the position in the current iteration.
divergence_tol: Divergence tolerance - solver aborts if
- `norm(constr(pos)) > divergence_tol` on any iteration where `pos` is the
- position at the current iteration and raises `mici.errors.ConvergenceError`.
+ :code:`norm(constr(pos)) > divergence_tol` on any iteration where
+ :code:`pos` is the position at the current iteration and raises
+ :py:exc:`mici.errors.ConvergenceError`.
max_iters: Maximum number of iterations to perform before aborting and raising
- `mici.errors.ConvergenceError`.
+ :py:exc:`mici.errors.ConvergenceError`.
norm: Norm to use to test for convergence.
Returns:
- Updated `state` object with position component satisfying constraint equation to
- within `constraint_tol`, i.e. `norm(system.constr(state.pos)) < constraint_tol`.
+ Updated state object with position component satisfying constraint equation to
+ within :code:`constraint_tol`, i.e.
+ :code:`norm(system.constr(state.pos)) < constraint_tol`.
Raises:
- mici.errors.ConvergenceError: If solver does not converge within `max_iters`
- iterations, diverges or encounters a `ValueError` during the iteration.
+ mici.errors.ConvergenceError: If solver does not converge within
+ :code:`max_iters` iterations, diverges or encounters a :py:exc:`ValueError`
+ during the iteration.
"""
mu = np.zeros_like(state.pos)
jacob_constr_prev = system.jacob_constr(state_prev)
# Use absolute value of dt and adjust for sign of dt in mom update below
- dh2_flow_pos_dmom, dh2_flow_mom_dmom = system.dh2_flow_dmom(abs(time_step))
+ dh2_flow_pos_dmom, dh2_flow_mom_dmom = system.dh2_flow_dmom(
+ state_prev, abs(time_step)
+ )
for i in range(max_iters):
try:
jacob_constr = system.jacob_constr(state)
@@ -492,7 +538,7 @@ Source code for mici.solvers
state: ChainState,
state_prev: ChainState,
time_step: float,
- system: System,
+ system: ConstrainedEuclideanMetricSystem,
constraint_tol: float = 1e-9,
position_tol: float = 1e-8,
divergence_tol: float = 1e10,
@@ -500,60 +546,70 @@ Source code for mici.solvers
max_line_search_iters: int = 10,
norm: ScalarFunction = maximum_norm,
) -> ChainState:
- """Solve constraint equation using Newton's method with backtracking line-search.
+ r"""Solve constraint equation using Newton's method with backtracking line-search.
- Requires re-evaluating both the constraint function `system.constr` and constraint
- Jacobian `system.jacob_constr` within the solver loop and computation of matrix
- decompositions on each iteration.
+ Requires re-evaluating both the constraint function :code:`system.constr` and
+ constraint Jacobian :code:`system.jacob_constr` within the solver loop and
+ computation of matrix decompositions on each iteration.
Solves an equation of the form
- .. code::
+ .. math::
+
+ r(\lambda) = c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda)) = 0,
- r(λ) = c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) = 0
+ for the vector of Lagrange multipliers :math:`\lambda` to project a point on to the
+ manifold defined by the zero level set of a constraint function :math:`c`, with
+ :math:`\Phi_{2,1}` the flow map for the :math:`h_2` Hamiltonian component for the
+ system restricted to the position component output.
- for the vector of Lagrange multipliers `λ` to project a point on to the manifold
- defined by the zero level set of a constraint function `c` (`system.constr`), with
- `Φ₂,₁` the position component of a flow map for the `h₂` Hamiltonian component for
- the system (i.e. `system.h2_flow`).
+ The Jacobian of the residual function :math:`r` is
- The Jacobian of the residual function `r` is
+ .. math::
- .. code::
+ \partial r(\lambda) =
+ \partial c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda))
+ \partial_2 (\Phi_{2,1}(t))(q, p + \partial c(q)^T \lambda)
+ \partial c(q)^T
- ∂r(λ) = ∂c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) ∂₂Φ₂,₁ ∂c(q)ᵀ
+ where :math:`\partial_2 (\Phi_{2,1}(t))` is the Jacobian of the (position
+ restricted) flow-map :math:`\Phi_{2,1}` with respect to its second (momentum)
+ argument. We assume here that :math:`\Phi_{2,1}(t)` is linear in its second
+ (momentum) argument, such that :math:`\partial_2 (\Phi_{2,1}(t))(q, p + \partial
+ c(q)^T \lambda)` is constant with respect to :math:`\lambda`.
- where `∂₂Φ₂,₁` is the Jacobian of the (linear) flow-map `Φ₂,₁` with respect to the
- second (momentum argument), such that the scaled Newton update is
+ The scaled Newton update is
- .. code::
+ .. math::
- λ_(α) = λ - α * ∂r(λ)⁻¹ r(λ)
+ \lambda'(\alpha) = \lambda - \alpha \partial r(\lambda)^{-1} r(\lambda)
- which requires evaluating `∂c` (`system.jacob_constr`) on each iteration and solving
- a linear system in the residual Jacobian `∂r(λ)`.
+ which requires evaluating :math:`\partial c` on each iteration and solving a linear
+ system in the residual Jacobian :math:`\partial r(\lambda)`.
- The step size `α ∈ [0, 1]` is initialised at `α = 1` with a backtracking line search
- performed by multiplying `α` by 0.5 until `|r(λ_(α))| < |r(λ)`.
+ The step size :math:`\alpha \in [0, 1]` is initialised at :math:`\alpha = 1` with a
+ backtracking line search performed by multiplying :math:`\alpha` by 0.5 until
+ :math:`|r(\lambda'(\alpha))| < |r(\lambda)`.
Args:
- state: Post `h2_flow` update state to project.
- state_prev: Previous state in co-tangent bundle before `h2_flow` update which
- defines the co-tangent space to perform projection in.
- time_step: Integrator time step used in `h2_flow` update.
- system: Hamiltonian system defining `h2_flow` and `constr` functions used to
- define constraint equation to solve.
+ state: Post :code:`h2_flow` update state to project.
+ state_prev: Previous state in co-tangent bundle before :code:`h2_flow` update
+ which defines the co-tangent space to perform projection in.
+ time_step: Integrator time step used in :code:`h2_flow` update.
+ system: Hamiltonian system defining :code:`h2_flow` and :code:`constr` functions
+ used to define constraint equation to solve.
constraint_tol: Convergence tolerance in constraint space. Iteration will
- continue until `norm(constr(pos)) < constraint_tol` where `pos` is the
- position at the current iteration.
+ continue until :code:`norm(constr(pos)) < constraint_tol` where :code:`pos`
+ is the position at the current iteration.
position_tol: Convergence tolerance in position space. Iteration will continue
- until `norm(delt_pos) < position_tol` where `delta_pos` is the change in the
- position in the current iteration.
+ until :code:`norm(delta_pos) < position_tol` where :code:`delta_pos` is the
+ change in the position in the current iteration.
divergence_tol: Divergence tolerance - solver aborts if
- `norm(constr(pos)) > divergence_tol` on any iteration where `pos` is the
- position at the current iteration and raises `mici.errors.ConvergenceError`.
+ :code:`norm(constr(pos)) > divergence_tol` on any iteration where
+ :code:`pos` is the position at the current iteration and raises
+ :py:exc:`mici.errors.ConvergenceError`.
max_iters: Maximum number of iterations to perform before aborting and raising
- `mici.errors.ConvergenceError`.
+ :py:exc:`mici.errors.ConvergenceError`.
max_line_search_iters: Maximum number of 'inner' line search iterations to
perform to try to find step size along search direction which decreases
residual norm.
@@ -561,16 +617,22 @@ Source code for mici.solvers
Returns:
Updated state object with position component satisfying constraint equation to
- within `constraint_tol`, i.e. `norm(system.constr(state.pos)) < constraint_tol`.
+ within :code:`constraint_tol`, i.e.
+ :code:`norm(system.constr(state.pos)) < constraint_tol`.
Raises:
- mici.errors.ConvergenceError: If solver does not converge within `max_iters`
- iterations, diverges or encounters a `ValueError` during the iteration.
+ mici.errors.ConvergenceError: If solver does not converge within
+ :code:`max_iters` iterations, diverges or encounters a :py:exc:`ValueError`
+ during the iteration.
"""
mu = np.zeros_like(state.pos)
jacob_constr_prev = system.jacob_constr(state_prev)
# Use absolute value of dt and adjust for sign of dt in mom update below
- dh2_flow_pos_dmom, dh2_flow_mom_dmom = system.dh2_flow_dmom(abs(time_step))
+ dh2_flow_pos_dmom, dh2_flow_mom_dmom = system.dh2_flow_dmom(
+ state_prev, abs(time_step)
+ )
+ # Initialize with dummy values to avoid undefined name linter errors
+ delta_pos, step_size = None, None
for i in range(max_iters):
try:
jacob_constr = system.jacob_constr(state)
diff --git a/_modules/mici/stagers.html b/_modules/mici/stagers.html
index 00b4df8..9bebbbd 100644
--- a/_modules/mici/stagers.html
+++ b/_modules/mici/stagers.html
@@ -75,10 +75,11 @@ Source code for mici.stagers
from __future__ import annotations
import abc
-from typing import NamedTuple, TYPE_CHECKING
+from typing import TYPE_CHECKING, NamedTuple
if TYPE_CHECKING:
from typing import Iterable, Optional
+
from mici.adapters import Adapter
from mici.types import TraceFunction
diff --git a/_modules/mici/states.html b/_modules/mici/states.html
index b1eccc2..ca381f1 100644
--- a/_modules/mici/states.html
+++ b/_modules/mici/states.html
@@ -75,14 +75,17 @@ Source code for mici.states
from __future__ import annotations
import copy
-from functools import wraps
from collections import Counter
+from functools import wraps
from typing import TYPE_CHECKING
+
from mici.errors import ReadOnlyStateError
if TYPE_CHECKING:
from typing import Any, Callable, Iterable, Optional
+
from numpy.typing import ArrayLike
+
from mici.systems import System
diff --git a/_modules/mici/systems.html b/_modules/mici/systems.html
index b6a3bc7..78ca24d 100644
--- a/_modules/mici/systems.html
+++ b/_modules/mici/systems.html
@@ -76,32 +76,26 @@ Source code for mici.systems
from abc import ABC, abstractmethod
from typing import TYPE_CHECKING
+
import numpy as np
-from mici.states import cache_in_state, cache_in_state_with_aux
+
import mici.matrices as matrices
from mici.autodiff import autodiff_fallback
+from mici.states import cache_in_state, cache_in_state_with_aux
if TYPE_CHECKING:
from typing import Any, Optional, Type
+
from numpy.random import Generator
+
from mici.states import ChainState
- from mici.types import (
- ScalarLike,
- ArrayLike,
- MatrixLike,
- MetricLike,
- ArrayFunction,
- ScalarFunction,
- GradientFunction,
- HessianFunction,
- MatrixTressianProduct,
- MatrixTressianProductFunction,
- JacobianFunction,
- MatrixHessianProduct,
- MatrixHessianProductFunction,
- VectorJacobianProduct,
- VectorJacobianProductFunction,
- )
+ from mici.types import (ArrayFunction, ArrayLike, GradientFunction,
+ HessianFunction, JacobianFunction,
+ MatrixHessianProduct, MatrixHessianProductFunction,
+ MatrixLike, MatrixTressianProduct,
+ MatrixTressianProductFunction, MetricLike,
+ ScalarFunction, ScalarLike, VectorJacobianProduct,
+ VectorJacobianProductFunction)
[docs]class System(ABC):
@@ -301,7 +295,7 @@ Source code for mici.systems
and :math:`h_1` and :math:`h_2` Hamiltonian component functions. The exact
Hamiltonian flows for both the :math:`h_1` and :math:`h_2` components are assumed to
be tractable for subclasses of this class.
-
+
By default :math:`h_1` is assumed to correspond to the negative logarithm of an
unnormalized density on the position variables with respect to a reference measure,
with the corresponding distribution on the position space being the target
@@ -317,22 +311,6 @@ Source code for mici.systems
Args:
state: State to start flow at.
dt: Time interval to simulate flow for.
- """
-
-[docs] @abstractmethod
- def dh2_flow_dmom(self, dt: ScalarLike) -> tuple[matrices.Matrix, matrices.Matrix]:
- """Derivatives of `h2_flow` flow map with respect to momentum argument.
-
- Args:
- dt: Time interval flow simulated for.
-
- Returns:
- Tuple `(dpos_dmom, dmom_dmom)` with :code:`dpos_dmom` a matrix representing
- derivative (Jacobian) of position output of :py:meth:`h2_flow` with respect
- to the value of the momentum component of the initial input state and
- :code:`dmom_dmom` a matrix representing derivative (Jacobian) of momentum
- output of :py:meth:`h2_flow` with respect to the value of the momentum
- component of the initial input state.
"""
@@ -425,9 +403,6 @@ Source code for mici.systems
-[docs] def dh2_flow_dmom(self, dt: ScalarLike) -> tuple[matrices.Matrix, matrices.Matrix]:
- return (dt * self.metric.inv, matrices.IdentityMatrix(self.metric.shape[0]))
-
[docs] def sample_momentum(self, state: ChainState, rng: Generator) -> ArrayLike:
return self.metric.sqrt @ rng.standard_normal(state.pos.shape)
@@ -517,7 +492,7 @@ Source code for mici.systems
return state.pos
[docs] def h2_flow(self, state: ChainState, dt: ScalarLike):
- omega = 1.0 / self.metric.eigval ** 0.5
+ omega = 1.0 / self.metric.eigval**0.5
sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt)
eigvec_T_pos = self.metric.eigvec.T @ state.pos
eigvec_T_mom = self.metric.eigvec.T @ state.mom
@@ -526,20 +501,151 @@ Source code for mici.systems
)
state.mom = self.metric.eigvec @ (
cos_omega_dt * eigvec_T_mom - (sin_omega_dt / omega) * eigvec_T_pos
- )
-
-[docs] def dh2_flow_dmom(self, dt: ScalarLike) -> tuple[matrices.Matrix, matrices.Matrix]:
- omega = 1.0 / self.metric.eigval ** 0.5
- sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt)
- return (
- matrices.EigendecomposedSymmetricMatrix(
- self.metric.eigvec, sin_omega_dt * omega
- ),
- matrices.EigendecomposedSymmetricMatrix(self.metric.eigvec, cos_omega_dt),
)
-[docs]class ConstrainedEuclideanMetricSystem(EuclideanMetricSystem):
+[docs]class ConstrainedTractableFlowSystem(TractableFlowSystem):
+ r"""Base class for Hamiltonian systems subject to constraints with tractable flows.
+
+ The (constrained) position space is assumed to be a differentiable manifold embedded
+ with a :math:`Q`-dimensional ambient Euclidean space. The :math:`Q-C` dimensional
+ manifold :math:`\mathcal{M}` is implicitly defined by an equation
+
+ .. math::
+
+ \mathcal{M} = \lbrace q \in \mathbb{R}^Q : c(q) = 0 \rbrace`
+
+ with :math:`c: \mathbb{R}^Q \to \mathbb{R}^C` the differentiable and surjective
+ vector-valued *constraint function*.
+
+ The Hamiltonian function :math:`h` is assumed to have the general form
+
+ .. math::
+
+ h(q, p) = h_1(q) + h_2(q, p)
+
+ where :math:`q` and :math:`p` are the position and momentum variables respectively,
+ and :math:`h_1` and :math:`h_2` Hamiltonian component functions. The exact
+ *unconstrained* Hamiltonian flows for both the :math:`h_1` and :math:`h_2`
+ components, respectively :math:`\Phi_1` and :math:`\Phi_2` are assumed to be
+ tractable for subclasses of this class.
+
+ The *constrained* Hamiltonian dynamics are described by the system of differential
+ algebraic equations
+
+ .. math::
+
+ \dot{q} = \nabla_2 h(q, p), \quad
+ \dot{p} = -\nabla_1 h(q, p) - \partial c(q)^T\lambda, \quad
+ c(q) = 0.
+
+ where math:`\lambda` is a set of Lagrange multipliers of dimension equal to number
+ of constraints, :math:`C`, and which are implicitly defined by the condition that
+ the constraint equation :math:`c(q) = 0` applies at all times.
+
+ The dynamics implicitly define a set of constraints on the momentum variables,
+ differentiating the constraint equation with respect to time giving that
+
+ .. math::
+
+ \partial c(q) \nabla_2 h(q, p) = \partial c(q) \nabla_2 h_2(q, p) = 0.
+
+ The set of momentum variables satisfying the above for given position variables is
+ termed the cotangent space of the manifold (at a position). Here we assume that the
+ operation of projecting a momentum vector onto the cotangent space at a given
+ position is tractable to compute.
+ """
+
+[docs] @abstractmethod
+ def constr(self, state: ChainState) -> ArrayLike:
+ """Constraint function at the current position.
+
+ Args:
+ state: State to compute value at.
+
+ Returns:
+ Value of :code:`constr(state.pos)` as 1D array.
+ """
+
+[docs] @abstractmethod
+ def jacob_constr(self, state: ChainState) -> ArrayLike:
+ """Jacobian of constraint function at the current position.
+
+ Args:
+ state: State to compute value at.
+
+ Returns:
+ Value of Jacobian of :code:`constr(state.pos)` as 2D array.
+ """
+
+[docs] @abstractmethod
+ def jacob_constr_inner_product(
+ self,
+ jacob_constr_1: MatrixLike,
+ inner_product_matrix: matrices.PositiveDefiniteMatrix,
+ jacob_constr_2: Optional[MatrixLike] = None,
+ ) -> MatrixLike:
+ """Compute inner product of rows of constraint Jacobian matrices.
+
+ Computes :code:`jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.T`
+ potentially exploiting any structure / sparsity in :code:`jacob_constr_1`,
+ :code:`jacob_constr_2` and :code:`inner_product_matrix`.
+
+ Args:
+ jacob_constr_1: First constraint Jacobian in product.
+ inner_product_matrix: Positive-definite matrix defining inner-product
+ between rows of two constraint Jacobians.
+ jacob_constr_2: Second constraint Jacobian in product. Defaults to
+ :code:`jacob_constr_1` if set to :code:`None`.
+
+ Returns
+ Object corresponding to computed inner products of the constraint Jacobian
+ rows.
+ """
+
+[docs] @abstractmethod
+ def dh2_flow_dmom(
+ self, state: ChainState, dt: ScalarLike
+ ) -> tuple[matrices.Matrix, matrices.Matrix]:
+ """Derivatives of :py:meth:`h2_flow` flow map with respect to momentum argument.
+
+ Args:
+ state: State to evaluate derivatives of flow map at.
+ dt: Time interval flow simulated for.
+
+ Returns:
+ Tuple :code:`(dpos_dmom, dmom_dmom)` with :code:`dpos_dmom` a matrix
+ representing derivative (Jacobian) of position output of :py:meth:`h2_flow`
+ with respect to the value of the momentum component of the initial input
+ state and :code:`dmom_dmom` a matrix representing derivative (Jacobian) of
+ momentum output of :py:meth:`h2_flow` with respect to the value of the
+ momentum component of the initial input state.
+ """
+
+[docs] @abstractmethod
+ def project_onto_cotangent_space(
+ self, mom: ArrayLike, state: ChainState
+ ) -> ArrayLike:
+ """Project a momentum on to the co-tangent space at a position.
+
+ Args:
+ mom: Momentum (co-)vector as 1D array to project on to co-tangent space.
+ state: State definining position on the manifold to project in to the
+ co-tangent space of.
+
+ Returns:
+ Projected momentum in the co-tangent space at :code:`state.pos`.
+ """
+
+[docs] def sample_momentum(self, state: ChainState, rng: Generator) -> ArrayLike:
+ mom = super().sample_momentum(state, rng)
+ mom = self.project_onto_cotangent_space(mom, state)
+ return mom
+
+
+[docs]class ConstrainedEuclideanMetricSystem(
+ ConstrainedTractableFlowSystem, EuclideanMetricSystem
+):
r"""Base class for Euclidean Hamiltonian systems subject to constraints.
The (constrained) position space is assumed to be a differentiable manifold embedded
@@ -559,7 +665,7 @@ Source code for mici.systems
h_2(q, p) = \frac{1}{2} p^T M^{-1} p.
The time-derivative of the constraint equation implies a further set of constraints
- on the momentum :math:`q` with :math:`\partial c(q) M^{-1} p = 0` at all time
+ on the momentum :math:`p` with :math:`\partial c(q) M^{-1} p = 0` at all time
points, corresponding to the momentum (velocity) being in the co-tangent space
(tangent space) to the manifold.
@@ -694,52 +800,16 @@ Source code for mici.systems
[docs] @cache_in_state("pos")
def constr(self, state: ChainState) -> ArrayLike:
- """Constraint function at the current position.
-
- Args:
- state: State to compute value at.
-
- Returns:
- Value of :code:`constr(state.pos)` as 1D array.
- """
return self._constr(state.pos)
[docs] @cache_in_state_with_aux("pos", "constr")
def jacob_constr(self, state: ChainState) -> ArrayLike:
- """Jacobian of constraint function at the current position.
-
- Args:
- state: State to compute value at.
-
- Returns:
- Value of Jacobian of :code:`constr(state.pos)` as 2D array.
- """
return self._jacob_constr(state.pos)
-[docs] @abstractmethod
- def jacob_constr_inner_product(
- self,
- jacob_constr_1: MatrixLike,
- inner_product_matrix: matrices.PositiveDefiniteMatrix,
- jacob_constr_2: Optional[MatrixLike] = None,
- ) -> MatrixLike:
- """Compute inner product of rows of constraint Jacobian matrices.
-
- Computes :code:`jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.T`
- potentially exploiting any structure / sparsity in :code:`jacob_constr_1`,
- :code:`jacob_constr_2` and :code:`inner_product_matrix`.
-
- Args:
- jacob_constr_1: First constraint Jacobian in product.
- inner_product_matrix: Positive-definite matrix defining inner-product
- between rows of two constraint Jacobians.
- jacob_constr_2: Second constraint Jacobian in product. Defaults to
- :code:`jacob_constr_1` if set to :code:`None`.
-
- Returns
- Object corresponding to computed inner products of the constraint Jacobian
- rows.
- """
+[docs] def dh2_flow_dmom(
+ self, state: ChainState, dt: ScalarLike
+ ) -> tuple[matrices.Matrix, matrices.Matrix]:
+ return (dt * self.metric.inv, matrices.IdentityMatrix(self.metric.shape[0]))
[docs] @cache_in_state("pos")
def gram(self, state: ChainState) -> matrices.PositiveDefiniteMatrix:
@@ -807,26 +877,11 @@ Source code for mici.systems
[docs] def project_onto_cotangent_space(
self, mom: ArrayLike, state: ChainState
) -> ArrayLike:
- """Project a momentum on to the co-tangent space at a position.
-
- Args:
- mom: Momentum (co-)vector as 1D array to project on to co-tangent space.
- state: State definining position on the manifold to project in to the
- co-tangent space of.
-
- Returns:
- Projected momentum in the co-tangent space at :code:`state.pos`.
- """
# Use parenthesis to force right-to-left evaluation to avoid
# matrix-matrix products
mom -= self.jacob_constr(state).T @ (
self.inv_gram(state) @ (self.jacob_constr(state) @ (self.metric.inv @ mom))
)
- return mom
-
-[docs] def sample_momentum(self, state: ChainState, rng: Generator) -> ArrayLike:
- mom = super().sample_momentum(state, rng)
- mom = self.project_onto_cotangent_space(mom, state)
return mom
@@ -980,10 +1035,10 @@ Source code for mici.systems
[docs]class GaussianDenseConstrainedEuclideanMetricSystem(
GaussianEuclideanMetricSystem, DenseConstrainedEuclideanMetricSystem
):
- r"""Gaussian Euclidean Hamiltonian system st. a dense set of constraints.
+ """Gaussian Euclidean Hamiltonian system subject to a dense set of constraints.
- See `ConstrainedEuclideanMetricSystem` for more details about constrained
- systems and `GaussianEuclideanMetricSystem` for Gaussian Euclidean metric
+ See :py:class:`ConstrainedEuclideanMetricSystem` for more details about constrained
+ systems and :py:class:`GaussianEuclideanMetricSystem` for Gaussian Euclidean metric
systems.
"""
@@ -1010,7 +1065,7 @@ Source code for mici.systems
manifold corresponding to the ratio of the prior density (specified by
`neg_log_dens`) and the square-root of the determinant of the Gram
matrix defined by
-
+
.. code-block::
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
@@ -1059,7 +1114,7 @@ Source code for mici.systems
*matrix-Hessian-product* (MHP) of the constraint function `constr` with
respect to the position array argument. The MHP is here defined as a
function of a `(dim_constr, dim_pos)` shaped 2D array `m`
-
+
.. code-block::
mhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1))
@@ -1104,7 +1159,19 @@ Source code for mici.systems
else:
return matrices.DenseSquareMatrix(
jacob_constr_1 @ (inner_product_matrix @ jacob_constr_2.T)
- )
+ )
+
+[docs] def dh2_flow_dmom(
+ self, state: ChainState, dt: ScalarLike
+ ) -> tuple[matrices.Matrix, matrices.Matrix]:
+ omega = 1.0 / self.metric.eigval**0.5
+ sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt)
+ return (
+ matrices.EigendecomposedSymmetricMatrix(
+ self.metric.eigvec, sin_omega_dt * omega
+ ),
+ matrices.EigendecomposedSymmetricMatrix(self.metric.eigvec, cos_omega_dt),
+ )
[docs]class RiemannianMetricSystem(System):
@@ -1581,7 +1648,7 @@ Source code for mici.systems
*vector-Jacobian-product* (VJP) of `metric_func` with respect to the
position array argument. The VJP is here defined as a function of a 2D
array `v`
-
+
.. code-block::
vjp(v) = sum(v[:, :, None] * jacob[:, :, :], axis=(0, 1))
@@ -1589,7 +1656,7 @@ Source code for mici.systems
where `jacob` is the `(dim_pos, dim_pos, dim_pos)` shaped Jacobian of `M
= metric_func(q)` with respect to `q` i.e. the array of partial
derivatives of the function such that
-
+
.. code-block::
jacob[i, j, k] = ∂M[i, j] / ∂q[k]
@@ -1636,7 +1703,7 @@ Source code for mici.systems
1D array of real eigenvalues, and `eigvec` the corresponding 2D array (orthogonal
matrix) with eigenvectors as columns, then the resulting positive-definite metric
matrix representation `M` is computed as
-
+
.. code-block::
M = eigvec @ diag(softabs(eigval, softabs_coeff)) @ eigvec.T
@@ -1695,7 +1762,7 @@ Source code for mici.systems
*matrix-Tressian-product* (MTP) of `neg_log_dens` with respect to the
position array argument. The MTP is here defined as a function of a
matrix `m` corresponding to
-
+
.. code-block::
mtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1))
@@ -1703,7 +1770,7 @@ Source code for mici.systems
where `tress` is the 'Tressian' of `f = neg_log_dens(q)` wrt `q` i.e.
the 3D array of third-order partial derivatives of the scalar-valued
function such that
-
+
.. code-block::
tress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k])
@@ -1767,7 +1834,7 @@ Source code for mici.systems
The matrix-Tressian-product (MTP) is here defined as a function of a matrix `m`
corresponding to
-
+
.. code-block::
mtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1))
@@ -1775,7 +1842,7 @@ Source code for mici.systems
where `tress` is the 'Tressian' of `f = neg_log_dens(q)` with respect to `q =
state.pos` i.e. the 3D array of third-order partial derivatives of the
scalar-valued function such that
-
+
.. code-block::
tress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k])
diff --git a/_modules/mici/transitions.html b/_modules/mici/transitions.html
index e0a12dd..7cb8546 100644
--- a/_modules/mici/transitions.html
+++ b/_modules/mici/transitions.html
@@ -74,23 +74,22 @@ Source code for mici.transitions
from __future__ import annotations
-from abc import ABC, abstractmethod, abstractproperty
import logging
-from typing import NamedTuple, TYPE_CHECKING
+from abc import ABC, abstractmethod, abstractproperty
+from typing import TYPE_CHECKING, NamedTuple
+
import numpy as np
+
+from mici.errors import (ConvergenceError, Error, HamiltonianDivergenceError,
+ IntegratorError, NonReversibleStepError)
from mici.utils import LogRepFloat
-from mici.errors import (
- Error,
- IntegratorError,
- NonReversibleStepError,
- ConvergenceError,
- HamiltonianDivergenceError,
-)
if TYPE_CHECKING:
from typing import Optional
+
from numpy.random import Generator
from numpy.typing import ArrayLike, DTypeLike
+
from mici.integrators import Integrator
from mici.states import ChainState
from mici.systems import System
@@ -256,7 +255,7 @@ Source code for mici.transitions
state.mom = self.system.sample_momentum(state, rng)
elif self.mom_resample_coeff != 0:
mom_ind = self.system.sample_momentum(state, rng)
- state.mom *= (1.0 - self.mom_resample_coeff ** 2) ** 0.5
+ state.mom *= (1.0 - self.mom_resample_coeff**2) ** 0.5
state.mom += self.mom_resample_coeff * mom_ind
return state, None
@@ -270,7 +269,7 @@ Source code for mici.transitions
"""
@property
- def state_variables(self) -> Set[str]:
+ def state_variables(self) -> set[str]:
return {"pos", "mom", "dir"}
@property
@@ -660,7 +659,9 @@ Source code for mici.transitions
@abstractmethod
def _weight_ratio(
- self, numerator: ScalarLike, denominator: ScalarLike,
+ self,
+ numerator: ScalarLike,
+ denominator: ScalarLike,
) -> ScalarLike:
pass
diff --git a/_modules/mici/utils.html b/_modules/mici/utils.html
index 7bd34e5..c29aaa8 100644
--- a/_modules/mici/utils.html
+++ b/_modules/mici/utils.html
@@ -74,9 +74,10 @@ Source code for mici.utils
from __future__ import annotations
+from math import exp, expm1, inf, log, log1p, nan
from typing import TYPE_CHECKING
+
import numpy as np
-from math import log, exp, log1p, expm1, inf, nan
try:
import xxhash
@@ -87,6 +88,7 @@ Source code for mici.utils
if TYPE_CHECKING:
from typing import Optional
+
from mici.types import ScalarLike
diff --git a/genindex.html b/genindex.html
index 9c5dd33..7a8a3e1 100644
--- a/genindex.html
+++ b/genindex.html
@@ -288,6 +288,8 @@ C
constr() (mici.systems.ConstrainedEuclideanMetricSystem method)
+ - (mici.systems.ConstrainedTractableFlowSystem method)
+
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
- (mici.systems.GaussianDenseConstrainedEuclideanMetricSystem method)
@@ -298,6 +300,8 @@
C
- ConstrainedEuclideanMetricSystem (class in mici.systems)
- ConstrainedLeapfrogIntegrator (class in mici.integrators)
+
+ - ConstrainedTractableFlowSystem (class in mici.systems)
- ConvergenceError
@@ -353,6 +357,8 @@ D
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -381,6 +387,8 @@ D
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -409,6 +417,8 @@ D
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -436,21 +446,19 @@ D
- dh2_flow_dmom() (mici.systems.ConstrainedEuclideanMetricSystem method)
- - (mici.systems.DenseConstrainedEuclideanMetricSystem method)
+
- (mici.systems.ConstrainedTractableFlowSystem method)
- - (mici.systems.EuclideanMetricSystem method)
+
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
- (mici.systems.GaussianDenseConstrainedEuclideanMetricSystem method)
-
- - (mici.systems.GaussianEuclideanMetricSystem method)
-
- - (mici.systems.TractableFlowSystem method)
- dh_dmom() (mici.systems.CholeskyFactoredRiemannianMetricSystem method)
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -481,6 +489,8 @@ D
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -813,6 +823,8 @@ G
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -887,6 +899,8 @@ H
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -915,6 +929,8 @@ H
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -943,6 +959,8 @@ H
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -973,6 +991,8 @@ H
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -1000,6 +1020,8 @@ H
- h2_flow() (mici.systems.ConstrainedEuclideanMetricSystem method)
+ - (mici.systems.ConstrainedTractableFlowSystem method)
+
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
- (mici.systems.EuclideanMetricSystem method)
@@ -1156,6 +1178,8 @@
J
- jacob_constr() (mici.systems.ConstrainedEuclideanMetricSystem method)
+ - (mici.systems.ConstrainedTractableFlowSystem method)
+
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
- (mici.systems.GaussianDenseConstrainedEuclideanMetricSystem method)
@@ -1166,6 +1190,8 @@
J
- jacob_constr_inner_product() (mici.systems.ConstrainedEuclideanMetricSystem method)
+ - (mici.systems.ConstrainedTractableFlowSystem method)
+
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
- (mici.systems.GaussianDenseConstrainedEuclideanMetricSystem method)
@@ -1557,6 +1583,8 @@
N
- (mici.systems.ConstrainedEuclideanMetricSystem method)
+
+ - (mici.systems.ConstrainedTractableFlowSystem method)
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
@@ -1644,6 +1672,8 @@ P
- project_onto_cotangent_space() (mici.systems.ConstrainedEuclideanMetricSystem method)
+ - (mici.systems.ConstrainedTractableFlowSystem method)
+
- (mici.systems.DenseConstrainedEuclideanMetricSystem method)
- (mici.systems.GaussianDenseConstrainedEuclideanMetricSystem method)
@@ -1745,6 +1775,8 @@
S
- - sample_pymc3_model() (in module mici.interop)
+
- sample_pymc_model() (in module mici.interop)
- sample_stan_model() (in module mici.interop)
diff --git a/mici.adapters.html b/mici.adapters.html
index 8a033f9..028b348 100644
--- a/mici.adapters.html
+++ b/mici.adapters.html
@@ -208,16 +208,17 @@
Bases: Adapter
Dual averaging integrator step size adapter.
Implementation of the dual algorithm step size adaptation algorithm described in
-[1], a modified version of the stochastic optimisation scheme of [2]. By default the
-adaptation is performed to control the accept_prob statistic of an integration
-transition to be close to a target value but the statistic adapted on can be altered
-by changing the adapt_stat_func.
+Hoffman and Gelman (2014), a modified version of the stochastic optimisation scheme
+of Nesterov (2009). By default the adaptation is performed to control the
+accept_stat statistic of an integration transition to be close to a target
+value but the statistic adapted on can be altered by changing the
+adapt_stat_func.
References
-Hoffman, M.D. and Gelman, A., 2014. The No-U-turn sampler: adaptively setting
+
Hoffman, M.D. and Gelman, A. (2014). The No-U-turn sampler: adaptively setting
path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research,
15(1), pp.1593-1623.
Nesterov, Y., 2009. Primal-dual subgradient methods for convex problems.
+
Nesterov, Y. (2009). Primal-dual subgradient methods for convex problems.
Mathematical programming 120(1), pp.221-259.
@@ -227,19 +228,19 @@
being controlled during adaptation.
adapt_stat_func (Optional[AdaptationStatisticFunction]) – Function which given a dictionary of transition statistics
outputs the value of the statistic to control during adaptation. By
-default this is set to a function which simply selects the ‘accept_stat’
-value in the statistics dictionary.
log_step_size_reg_target (Optional[float]) – Value to regularize the controlled output
-(logarithm of the integrator step size) towards. If None set to
-log(10 * init_step_size) where init_step_size is the initial
-‘reasonable’ step size found by a coarse search as recommended in
-Hoffman and Gelman (2014). This has the effect of giving the dual
+(logarithm of the integrator step size) towards. If None set to
+log(10 * init_step_size) where init_step_size is the
+initial ‘reasonable’ step size found by a coarse search as recommended
+in Hoffman and Gelman (2014). This has the effect of giving the dual
averaging algorithm a tendency towards testing step sizes larger than
the initial value, with typically integrating with a larger step size
having a lower computational cost.
log_step_size_reg_coefficient (float) – Coefficient controlling amount of
regularisation of controlled output (logarithm of the integrator step
-size) towards log_step_size_reg_target. Defaults to 0.05 as
+size) towards log_step_size_reg_target. Defaults to 0.05 as
recommended in Hoffman and Gelman (2014).
iter_decay_coeff (float) – Coefficient controlling exponent of decay in schedule
weighting stochastic updates to smoothed log step size estimate. Should
@@ -253,14 +254,15 @@
> 0 has the effect of stabilising early iterations. Defaults to the
value of 10 as recommended in Hoffman and Gelman (2014).
max_init_step_size_iters (int) – Maximum number of iterations to use in initial
-search for a reasonable step size with an AdaptationError exception
-raised if a suitable step size is not found within this many iterations.
mici.errors.AdaptationError exception raised if a suitable
+step size is not found within this many iterations.
log_step_size_reducer (Optional[ReducerFunction]) – Reduction to apply to final per-chain step sizes
estimates to produce overall integrator step size for main chain stages.
The specified function should accept a sequence of logarithms of
estimated step sizes and output a non-negative step size to use. If
-None, the default, a function which computes the arithmetic mean of
-the step sizes is used.
None, the default, a function which computes the arithmetic mean
+of the per-chain step sizes is used.
@@ -348,29 +350,29 @@
class mici.adapters.OnlineCovarianceMetricAdapter(reg_iter_offset=5, reg_scale=0.001)[source]
Bases: Adapter
Dense metric adapter using online covariance estimates.
-Uses Welford’s algorithm [1] to stably compute an online estimate of the sample
-covariance matrix of the chain state position components during sampling. If online
-estimates are available from multiple independent chains, the final covariance
-matrix estimate is calculated from the per-chain statistics using a covariance
-variant due to Schubert and Gertz [2] of the parallel / batched incremental variance
-algorithm described by Chan et al. [3]. The covariance matrix estimates are
-optionally regularized towards a scaled identity matrix, with increasing weight for
-small number of samples, to decrease the effect of noisy estimates for small sample
-sizes, following the approach in Stan [4]. The metric matrix representation is set
-to a dense positive definite matrix corresponding to the inverse of the
-(regularized) covariance matrix estimate.
+Uses Welford’s algorithm (Welford, 1962) to stably compute an online estimate of the
+sample covariance matrix of the chain state position components during sampling. If
+online estimates are available from multiple independent chains, the final
+covariance matrix estimate is calculated from the per-chain statistics using a
+covariance variant due to Schubert and Gertz (2018) of the parallel / batched
+incremental variance algorithm described by Chan et al. (1979). The covariance
+matrix estimates are optionally regularized towards a scaled identity matrix, with
+increasing weight for small number of samples, to decrease the effect of noisy
+estimates for small sample sizes, following the approach in Stan (Carpenter et al.,
+2017). The metric matrix representation is set to a dense positive definite matrix
+corresponding to the inverse of the (regularized) covariance matrix estimate.
References
-Welford, B. P., 1962. Note on a method for calculating corrected sums of
+
Welford, B. P. (1962). Note on a method for calculating corrected sums of
squares and products. Technometrics, 4(3), pp. 419–420.
Schubert, E. and Gertz, M., 2018. Numerically stable parallel computation of
+
Schubert, E. and Gertz, M. (2018). Numerically stable parallel computation of
(co-)variance. ACM. p. 10. doi:10.1145/3221269.3223036.
Chan, T. F., Golub, G. H., LeVeque, R. J., 1979. Updating formulae and a
+
Chan, T. F., Golub, G. H. and LeVeque, R. J. (1979). Updating formulae and a
pairwise algorithm for computing sample variances. Technical Report
STAN-CS-79-773, Department of Computer Science, Stanford University.
Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt,
-M., Brubaker, M., Guo, J., Li, P. and Riddell, A., 2017. Stan: A probabilistic
-programming language. Journal of Statistical Software, 76(1).
- Parameters:
@@ -468,26 +470,26 @@
class mici.adapters.OnlineVarianceMetricAdapter(reg_iter_offset=5, reg_scale=0.001)[source]
Bases: Adapter
Diagonal metric adapter using online variance estimates.
-Uses Welford’s algorithm [1] to stably compute an online estimate of the sample
-variances of the chain state position components during sampling. If online
+
Uses Welford’s algorithm (Welford, 1962) to stably compute an online estimate of the
+sample variances of the chain state position components during sampling. If online
estimates are available from multiple independent chains, the final variance
estimate is calculated from the per-chain statistics using the parallel / batched
-incremental variance algorithm described by Chan et al. [2]. The variance estimates
-are optionally regularized towards a common scalar value, with increasing weight for
-small number of samples, to decrease the effect of noisy estimates for small sample
-sizes, following the approach in Stan [3]. The metric matrix representation is set
-to a diagonal matrix with diagonal elements corresponding to the reciprocal of the
-(regularized) variance estimates.
+incremental variance algorithm described by Chan et al. (1979). The variance
+estimates are optionally regularized towards a common scalar value, with increasing
+weight for small number of samples, to decrease the effect of noisy estimates for
+small sample sizes, following the approach in Stan (Carpenter et al., 2017). The
+metric matrix representation is set to a diagonal matrix with diagonal elements
+corresponding to the reciprocal of the (regularized) variance estimates.
References
-Welford, B. P., 1962. Note on a method for calculating corrected sums of
+
Welford, B. P. (1962). Note on a method for calculating corrected sums of
squares and products. Technometrics, 4(3), pp. 419–420.
Chan, T. F., Golub, G. H., LeVeque, R. J., 1979. Updating formulae and a
+
Chan, T. F., Golub, G. H. and LeVeque, R. J. (1979). Updating formulae and a
pairwise algorithm for computing sample variances. Technical Report
STAN-CS-79-773, Department of Computer Science, Stanford University.
Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt,
-M., Brubaker, M., Guo, J., Li, P. and Riddell, A., 2017. Stan: A probabilistic
-programming language. Journal of Statistical Software, 76(1).
- Parameters:
@@ -602,7 +604,7 @@
-
mici.adapters.default_adapt_stat_func(stats)[source]
-Function to extract default acceptance statistic used for step-size adaptation.
+Function to extract default statistic used for step-size adaptation.
- Parameters:
stats (TransitionStatistics) – Dictionary of transition statistics.
@@ -640,7 +642,7 @@
Compute minimum of step sizes from their logs.
- Parameters:
-log_step_sizes (Iterable[float]) – Logarithms of per-chain estimated step sizes.
+log_step_sizes (Collection[float]) – Logarithms of per-chain estimated step sizes.
- Return type:
-
diff --git a/mici.html b/mici.html
index f11e3f8..b71057a 100644
--- a/mici.html
+++ b/mici.html
@@ -149,7 +149,7 @@
Submodulesmici.interop module
@@ -244,6 +244,7 @@ Submodulesmici.systems module
CholeskyFactoredRiemannianMetricSystemConstrainedEuclideanMetricSystemConstrainedTractableFlowSystemDenseConstrainedEuclideanMetricSystemDenseRiemannianMetricSystemDiagonalRiemannianMetricSystem
-
-class mici.integrators.ConstrainedLeapfrogIntegrator(system, step_size=None, n_inner_step=1, reverse_check_tol=2e-08, reverse_check_norm=<function maximum_norm>, projection_solver=<function solve_projection_onto_manifold_quasi_newton>, projection_solver_kwargs=None)[source]
+class mici.integrators.ConstrainedLeapfrogIntegrator(system, step_size=None, n_inner_step=1, reverse_check_tol=2e-08, reverse_check_norm=<function maximum_norm>, projection_solver=<function solve_projection_onto_manifold_newton>, projection_solver_kwargs=None)[source]
Bases: TractableFlowIntegrator
Leapfrog integrator for constrained Hamiltonian systems.
The Hamiltonian function is assumed to be expressible as the sum of two components
@@ -254,7 +254,7 @@
simulated. Specifically it is assumed that the Hamiltonian function h takes the
form
-\[h(q, p) = h_1(q) + h_2(q, p)\]
+\[h(q, p) = h_1(q) + h_2(q, p),\]
where \(q\) and \(p\) are the position and momentum variables respectively,
and \(h_1\) and \(h_2\) Hamiltonian component functions for which the exact
flows, respectively \(\Phi_1\) and \(\Phi_2\), can be computed.
@@ -279,7 +279,7 @@
The dynamics implicitly define a set of constraints on the momentum variables,
differentiating the constraint equation with respect to time giving that
-\[\partial c(q) \nabla_2 h(q, p) = \nabla_2 h_2(q, p) = 0.\]
+\[\partial c(q) \nabla_2 h(q, p) = \partial c(q) \nabla_2 h_2(q, p) = 0.\]
The set of momentum variables satisfying the above for given position variables is
termed the cotangent space of the manifold (at a position), and the set of
position-momentum pairs for which the position is on the constraint manifold and the
@@ -319,7 +319,8 @@
\(\partial c(q') \nabla_2 h(q', p') = 0\).
This can be decomposed as first solving for \(\lambda\) such that
-\[c((\Phi_2(t) \circ \Pi(\lambda)(q, p))_1) = 0\]
+\[c((\Phi_2(t) \circ \Pi(\lambda)(q, p))_1)
+= c((\Phi_2(t)(q, p + \partial c(q)^T \lambda))_1) = 0,\]
i.e. solving for the values of the Lagrange multipliers such that the position
component of the output of \(\Phi_2(t) \circ \Pi(\lambda)\) is on the manifold,
with this typically a non-linear system of equations that will need to be solved
@@ -327,26 +328,54 @@
\circ \Pi(\lambda)\) is then projected in to the cotangent space to compute the final
state pair, with this projection step as noted above typically having an analytic
solution.
+For more details see Reich (1996) and section 7.5.1 in Leimkuhler and Reich (2004).
The overall second-order integrator is then defined as the symmetric composition
\[\Psi(t) = A(t / 2) \circ B(t / N)^N \circ A(t / 2),\]
where \(N\) is a positive integer corresponding to the number of ‘inner’
-\(h_2\) flow steps. This integrator exactly preserves the constraints at all
-steps, such that if an initial position momentum pair \((q, p)\) are in the
-cotangent bundle, the corresponding pair after calling the step() method of
-the integrator will also be in the cotangent bundle.
-For more details see Reich (1996) and section 7.5.1 in Leimkuhler and Reich (2004).
+\(h_2\) flow steps, following the ‘geodesic integrator’ formulation proposed by
+Leimkuhler and Matthews (2016). The additional flexibility introduced by having
+the possibility of \(N > 1\) is particularly of use when evaluation of
+\(\Phi_1\) is significantly more expensive than evaluation of \(\Phi_2\); in
+this case using \(N > 1\) can allow a larger time step \(t\) to be used than
+may be otherwise possible due to the need to ensure the iterative solver used in
+\(B\) does not diverge, with a smaller step size \(t / N\) used for the
+steps involving the iterative solves with the (cheaper) \(\Phi_2\) flow map
+and a larger step size \(t\) used for the steps involving the (more expensive)
+\(\Phi_1\) flow map.
+This integrator exactly preserves the constraints at all steps, such that if an
+initial position momentum pair \((q, p)\) are in the cotangent bundle, the
+corresponding pair after calling the step() method of the integrator will
+also be in the cotangent bundle, providing the iterative solver converges.
+As the iterative solves may fail to converge, or may converge to one of multiple
+solutions, following the approach proposed by Zappa, Holmes-Cerfon and Goodman
+(2018), an explicit reversibility check is performed to ensure the overall
+integrator step is time-reversible; see also Lelievre, Rousset and Stoltz (2019) for
+an analysis of this approach specifically in the context of Hamiltonian Monte Carlo.
+If the reversibility check fails or the iterative solver fails to converge an
+appropriate error is raised (mici.errors.NonReversibleStepError and
+mici.errors.ConvergenceError respectively).
References
Reich, S. (1996). Symplectic integration of constrained Hamiltonian systems by
composition methods. SIAM journal on numerical analysis, 33(2), 475-491.
Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
Cambridge University Press.
Leimkuhler, B., & Matthews, C. (2016). Efficient molecular dynamics using
+geodesic integration and solvent–solute splitting. Proceedings of the Royal
+Society A: Mathematical, Physical and Engineering Sciences, 472(2189),
+20160138.
Zappa, E., Holmes‐Cerfon, M., & Goodman, J. (2018). Monte Carlo on manifolds:
+sampling densities and integrating functions. Communications on Pure and
+Applied Mathematics, 71(12), 2609-2647.
Lelievre, T., Rousset, M., & Stoltz, G. (2019). Hybrid Monte Carlo methods for
+sampling probability measures on submanifolds. Numerische Mathematik, 143,
+379-421.
- Parameters:
-system (System) – Hamiltonian system to integrate the dynamics of.
system (ConstrainedTractableFlowSystem) – Hamiltonian system to integrate the constrained dynamics of.
step_size (Optional[float]) – Integrator time step. If set to None it is assumed that a
step size adapter will be used to set the step size before calling the
step() method.
system.h2_flow map
on the state momentum component with respect to the position component.
This is used to project the state position component back on to the
-manifold after an unconstrained system.h2_flow update.
+manifold after an unconstrained system.h2_flow update. If the
+solver fails to convege a mici.errors.ConvergenceError
+exception is raised.
projection_solver_kwargs (Optional[dict[str, Any]]) – Dictionary of any keyword arguments to
projection_solver.
-Fixed-point iterations are used to solve the non-linear systems of equations in the
-implicit component updates (\(B\) and \(C^*\)).
The resulting implicit integrator is a symmetric second-order method corresponding
to a symplectic partitioned Runge-Kutta method. See Section 6.3.2 in Leimkuhler and
Reich (2004) for more details.
+Fixed-point iterations are used to solve the non-linear systems of equations in the
+implicit component updates (\(B\) and \(C^*\)). As the iterative solves may
+fail to converge, or may converge to one of multiple solutions, following the
+approach proposed by Zappa, Holmes-Cerfon and Goodman (2018), an explicit
+reversibility check is performed to ensure the overall integrator step is
+time-reversible. If the reversibility check fails or the iterative solver fails to
+converge an appropriate error is raised
+(mici.errors.NonReversibleStepError and
+mici.errors.ConvergenceError respectively).
References
Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
Cambridge University Press.
Zappa, E., Holmes‐Cerfon, M., & Goodman, J. (2018). Monte Carlo on manifolds:
+sampling densities and integrating functions. Communications on Pure and
+Applied Mathematics, 71(12), 2609-2647.
- Parameters:
@@ -521,15 +562,24 @@
p' = p - t \nabla_1 h(q', p'))
\rbrace, \\
A^*(t)(q, p) = (q + t \nabla_2 h(q, p), p - t \nabla_1 h(q, p)).\end{split}\]
-A fixed-point iteration is used to solve the non-linear system of equations in the
-implicit Euler step \(A\).
The resulting implicit integrator is a second-order method corresponding to a
symplectic one-stage Runge-Kutta method. See Sections 4.1 and 6.3.1 in Leimkuhler
and Reich (2004) for more details.
+A fixed-point iteration is used to solve the non-linear system of equations in the
+implicit Euler step \(A\). As the iterative solve may fail to converge, or may
+converge to one of multiple solutions, following the approach proposed by
+Zappa, Holmes-Cerfon and Goodman (2018), an explicit reversibility check is
+performed to ensure the overall integrator step is time-reversible. If the
+reversibility check fails or the iterative solver fails to converge an appropriate
+error is raised (mici.errors.NonReversibleStepError and
+mici.errors.ConvergenceError respectively).
References
Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
Cambridge University Press.
Zappa, E., Holmes‐Cerfon, M., & Goodman, J. (2018). Monte Carlo on manifolds:
+sampling densities and integrating functions. Communications on Pure and
+Applied Mathematics, 71(12), 2609-2647.
- Parameters:
diff --git a/mici.interop.html b/mici.interop.html
index 815012e..a9bdee1 100644
--- a/mici.interop.html
+++ b/mici.interop.html
@@ -54,7 +54,7 @@
mici.integrators module
mici.interop module
@@ -105,7 +105,7 @@
-
mici.interop.convert_to_inference_data(traces, stats, energy_key='energy', lp_key='lp')[source]
-Convert Mici sample_chains output to arviz.InferenceData object.
+Convert Mici sample_chains output to arviz.InferenceData.
- Parameters:
@@ -140,12 +140,12 @@
--
-mici.interop.sample_pymc3_model(draws=1000, *, tune=1000, chains=None, cores=None, random_seed=None, progressbar=True, init='auto', jitter_max_retries=10, trace=None, return_inferencedata=False, model=None, target_accept=0.8, max_treedepth=10)[source]
-Generate approximate samples from posterior defined by a PyMC3 model.
+-
+mici.interop.sample_pymc_model(draws=1000, *, tune=1000, chains=None, cores=None, random_seed=None, progressbar=True, init='auto', jitter_max_retries=10, trace=None, return_inferencedata=False, model=None, target_accept=0.8, max_treedepth=10)[source]
+Generate approximate samples from posterior defined by a PyMC model.
Uses dynamic multinomial HMC algorithm in Mici with adaptive warm-up phase.
-This function replicates the interface of the pymc3.sampling.sample()
-function to allow using as a (partial) drop-in replacement.
+This function replicates the interface of the pymc.sample() function to
+allow using as a (partial) drop-in replacement.
- Parameters:
@@ -185,7 +185,7 @@
return_inferencedata (bool) – Whether to return the traces as an
arviz.InferenceData (True) object or a
dict (False).
model (Optional[Model]) – PyMC3 model defining posterior distribution to sample from. May be
+
model (Optional[pymc.Model]) – PyMC model defining posterior distribution to sample from. May be
None if function is called from within model context manager.
target_accept (float) – Target value for the acceptance statistic being controlled during
adaptive warm-up.
diff --git a/mici.solvers.html b/mici.solvers.html
index 6d7a149..3bc471b 100644
--- a/mici.solvers.html
+++ b/mici.solvers.html
@@ -16,6 +16,7 @@
+
@@ -112,7 +113,7 @@
-
class mici.solvers.FixedPointSolver(*args, **kwargs)[source]
Bases: Protocol
-Solver for fixed point equation, returning x such that func(x) = x.
+Solver for fixed point equation func(x) = x.
-
__call__(func, x0, **kwargs)[source]
@@ -141,13 +142,13 @@
Bases: Protocol
Solver for projection on to manifold step in constrained integrator.
Solves an equation of the form
-r(λ) = c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) = 0
-
-
-for the vector of Lagrange multipliers λ to project a point on to the manifold
-defined by the zero level set of a constraint function c (system.constr), with
-Φ₂,₁ the position component of a flow map for the h₂ Hamiltonian component for
-the system (i.e. system.h2_flow).
+
+\[r(\lambda) = c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda)) = 0,\]
+for the vector of Lagrange multipliers \(\lambda\) to project a point on to the
+manifold defined by the zero level set of a constraint function \(c\), with
+\(\Phi_{2,1}\) the flow map for the \(h_2\) Hamiltonian component for the
+system restricted to the position component output. The map \(\Phi_{2,1}\) is
+assumed to be linear in its second (momentum) argument.
state (ChainState) – Current chain state after unconstrained step.
state_prev (ChainState) – Previous chain state on manifold.
time_step (float) – Integrator time step for unconstrained step.
system (System) – Hamiltonian system dynamics are being simulated for.
system (ConstrainedTractableFlowSystem) – Hamiltonian system constrained dynamics are being simulated for.
- Returns:
@@ -187,26 +188,28 @@
-
mici.solvers.solve_fixed_point_direct(func, x0, convergence_tol=1e-09, divergence_tol=10000000000.0, max_iters=100, norm=<function maximum_norm>)[source]
-Solve fixed point equation func(x) = x using direct iteration.
+Solve fixed point equation func(x) = x using direct iteration.
- Parameters:
func (ArrayFunction) – Function to find fixed point of.
x0 (ArrayLike) – Initial state (function argument).
convergence_tol (float) – Convergence tolerance - solver successfully terminates when
-norm(func(x) - x) < convergence_tol.
norm(func(x) - x) < convergence_tol.
divergence_tol (float) – Divergence tolerance - solver aborts if
-norm(func(x) - x) > divergence_tol on any iteration.
norm(func(x) - x) > divergence_tol on any iteration.
max_iters (int) – Maximum number of iterations before raising exception.
norm (ScalarFunction) – Norm to use to assess convergence.
- Returns:
-Solution to fixed point equation with norm(func(x) - x) < convergence_tol.
+Solution to fixed point equation with
+norm(func(x) - x) < convergence_tol.
- Raises:
-ConvergenceError – If solver does not converge within max_iters
- iterations, diverges or encounters a ValueError during the iteration.
+ConvergenceError – If solver does not converge within
+ max_iters iterations, diverges or encounters a ValueError
+ during the iteration.
- Return type:
ArrayLike
@@ -217,30 +220,31 @@
-
mici.solvers.solve_fixed_point_steffensen(func, x0, convergence_tol=1e-09, divergence_tol=10000000000.0, max_iters=100, norm=<function maximum_norm>)[source]
-Solve fixed point equation func(x) = x using Steffensen’s method.
-Steffennsen’s method [1] achieves quadratic convergence but at the cost of two
-function evaluations per iteration so for functions where convergence is achieved in
-a small number of iterations, direct iteration may be cheaper.
-[1] : https://en.wikipedia.org/wiki/Steffensen%27s_method
+Solve fixed point equation func(x) = x using Steffensen’s method.
+Steffennsen’s method achieves quadratic convergence but at the cost of two function
+evaluations per iteration so for functions where convergence is achieved in a small
+number of iterations, direct iteration may be cheaper.
- Parameters:
func (ArrayFunction) – Function to find fixed point of.
x0 (ArrayLike) – Initial state (function argument).
convergence_tol (float) – Convergence tolerance - solver successfully terminates
-when norm(func(x) - x) < convergence_tol.
norm(func(x) - x) < convergence_tol.
divergence_tol (float) – Divergence tolerance - solver aborts if
-norm(func(x) - x) > divergence_tol on any iteration.
norm(func(x) - x) > divergence_tol on any iteration.
max_iters (int) – Maximum number of iterations before raising exception.
norm (ScalarFunction) – Norm to use to assess convergence.
- Returns:
-Solution to fixed point equation with norm(func(x) - x) < convergence_tol.
+Solution to fixed point equation with
+norm(func(x) - x) < convergence_tol.
- Raises:
-ConvergenceError – If solver does not converge within max_iters
- iterations, diverges or encounters a ValueError during the iteration.
+ConvergenceError – If solver does not converge within
+ max_iters iterations, diverges or encounters a ValueError
+ during the iteration.
- Return type:
ArrayLike
@@ -252,58 +256,65 @@
-
mici.solvers.solve_projection_onto_manifold_newton(state, state_prev, time_step, system, constraint_tol=1e-09, position_tol=1e-08, divergence_tol=10000000000.0, max_iters=50, norm=<function maximum_norm>)[source]
Solve constraint equation using Newton’s method.
-Requires re-evaluating both the constraint function system.constr and constraint
-Jacobian system.jacob_constr within the solver loop and computation of matrix
-decompositions on each iteration.
+Requires re-evaluating both the constraint function system.constr and
+constraint Jacobian system.jacob_constr within the solver loop and
+computation of matrix decompositions on each iteration.
Solves an equation of the form
-r(λ) = c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) = 0
-
-
-for the vector of Lagrange multipliers λ to project a point on to the manifold
-defined by the zero level set of a constraint function c (system.constr), with
-Φ₂,₁ the position component of a flow map for the h₂ Hamiltonian component for
-the system (i.e. system.h2_flow).
-The Jacobian of the residual function r is
-∂r(λ) = ∂c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) ∂₂Φ₂,₁ ∂c(q)ᵀ
-
-
-where ∂₂Φ₂,₁ is the Jacobian of the (linear) flow-map Φ₂,₁ with respect to the
-second (momentum argument), such that the Newton update is
-λ = λ - ∂r(λ)⁻¹ r(λ)
-
-
-which requires evaluating ∂c (system.jacob_constr) on each iteration and solving
-a linear system in the residual Jacobian ∂r(λ).
+
+\[r(\lambda) = c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda)) = 0,\]
+for the vector of Lagrange multipliers \(\lambda\) to project a point on to the
+manifold defined by the zero level set of a constraint function \(c\), with
+\(\Phi_{2,1}\) the flow map for the \(h_2\) Hamiltonian component for the
+system restricted to the position component output.
+The Jacobian of the residual function \(r\) is
+
+\[\partial r(\lambda) =
+\partial c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda))
+\partial_2 (\Phi_{2,1}(t))(q, p + \partial c(q)^T \lambda)
+\partial c(q)^T\]
+where \(\partial_2 (\Phi_{2,1}(t))\) is the Jacobian of the (position
+restricted) flow-map \(\Phi_{2,1}\) with respect to its second (momentum)
+argument. We assume here that \(\Phi_{2,1}(t)\) is linear in its second
+(momentum) argument, such that \(\partial_2 (\Phi_{2,1}(t))(q, p + \partial
+c(q)^T \lambda)\) is constant with respect to \(\lambda\).
+The full Newton update is
+
+\[\lambda' = \lambda - \partial r(\lambda)^{-1} r(\lambda)\]
+which requires evaluating \(\partial c\) on each iteration and solving a linear
+system in the residual Jacobian \(\partial r(\lambda)\).
- Parameters:
-state (ChainState) – Post h2_flow update state to project.
state_prev (ChainState) – Previous state in co-tangent bundle before h2_flow update which
-defines the co-tangent space to perform projection in.
time_step (float) – Integrator time step used in h2_flow update.
system (System) – Hamiltonian system defining h2_flow and constr functions used to
-define constraint equation to solve.
state (ChainState) – Post h2_flow update state to project.
state_prev (ChainState) – Previous state in co-tangent bundle before h2_flow update
+which defines the co-tangent space to perform projection in.
time_step (float) – Integrator time step used in h2_flow update.
system (ConstrainedEuclideanMetricSystem) – Hamiltonian system defining h2_flow and constr functions
+used to define constraint equation to solve.
constraint_tol (float) – Convergence tolerance in constraint space. Iteration will
-continue until norm(constr(pos)) < constraint_tol where pos is the
-position at the current iteration.
norm(constr(pos)) < constraint_tol where pos
+is the position at the current iteration.
position_tol (float) – Convergence tolerance in position space. Iteration will continue
-until norm(delt_pos) < position_tol where delta_pos is the change in the
-position in the current iteration.
norm(delta_pos) < position_tol where delta_pos is the
+change in the position in the current iteration.
divergence_tol (float) – Divergence tolerance - solver aborts if
-norm(constr(pos)) > divergence_tol on any iteration where pos is the
-position at the current iteration and raises mici.errors.ConvergenceError.
norm(constr(pos)) > divergence_tol on any iteration where
+pos is the position at the current iteration and raises
+mici.errors.ConvergenceError.
max_iters (int) – Maximum number of iterations to perform before aborting and raising
-mici.errors.ConvergenceError.
mici.errors.ConvergenceError.
norm (ScalarFunction) – Norm to use to test for convergence.
- Returns:
-Updated state object with position component satisfying constraint equation to
-within constraint_tol, i.e. norm(system.constr(state.pos)) < constraint_tol.
+Updated state object with position component satisfying constraint equation to
+within constraint_tol, i.e.
+norm(system.constr(state.pos)) < constraint_tol.
- Raises:
-ConvergenceError – If solver does not converge within max_iters
- iterations, diverges or encounters a ValueError during the iteration.
+ConvergenceError – If solver does not converge within
+ max_iters iterations, diverges or encounters a ValueError
+ during the iteration.
- Return type:
-
@@ -315,50 +326,56 @@
-
mici.solvers.solve_projection_onto_manifold_newton_with_line_search(state, state_prev, time_step, system, constraint_tol=1e-09, position_tol=1e-08, divergence_tol=10000000000.0, max_iters=50, max_line_search_iters=10, norm=<function maximum_norm>)[source]
Solve constraint equation using Newton’s method with backtracking line-search.
-Requires re-evaluating both the constraint function system.constr and constraint
-Jacobian system.jacob_constr within the solver loop and computation of matrix
-decompositions on each iteration.
+Requires re-evaluating both the constraint function system.constr and
+constraint Jacobian system.jacob_constr within the solver loop and
+computation of matrix decompositions on each iteration.
Solves an equation of the form
-r(λ) = c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) = 0
-
-
-for the vector of Lagrange multipliers λ to project a point on to the manifold
-defined by the zero level set of a constraint function c (system.constr), with
-Φ₂,₁ the position component of a flow map for the h₂ Hamiltonian component for
-the system (i.e. system.h2_flow).
-The Jacobian of the residual function r is
-∂r(λ) = ∂c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) ∂₂Φ₂,₁ ∂c(q)ᵀ
-
-
-where ∂₂Φ₂,₁ is the Jacobian of the (linear) flow-map Φ₂,₁ with respect to the
-second (momentum argument), such that the scaled Newton update is
-λ_(α) = λ - α * ∂r(λ)⁻¹ r(λ)
-
-
-which requires evaluating ∂c (system.jacob_constr) on each iteration and solving
-a linear system in the residual Jacobian ∂r(λ).
-The step size α ∈ [0, 1] is initialised at α = 1 with a backtracking line search
-performed by multiplying α by 0.5 until |r(λ_(α))| < |r(λ).
+
+\[r(\lambda) = c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda)) = 0,\]
+for the vector of Lagrange multipliers \(\lambda\) to project a point on to the
+manifold defined by the zero level set of a constraint function \(c\), with
+\(\Phi_{2,1}\) the flow map for the \(h_2\) Hamiltonian component for the
+system restricted to the position component output.
+The Jacobian of the residual function \(r\) is
+
+\[\partial r(\lambda) =
+\partial c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda))
+\partial_2 (\Phi_{2,1}(t))(q, p + \partial c(q)^T \lambda)
+\partial c(q)^T\]
+where \(\partial_2 (\Phi_{2,1}(t))\) is the Jacobian of the (position
+restricted) flow-map \(\Phi_{2,1}\) with respect to its second (momentum)
+argument. We assume here that \(\Phi_{2,1}(t)\) is linear in its second
+(momentum) argument, such that \(\partial_2 (\Phi_{2,1}(t))(q, p + \partial
+c(q)^T \lambda)\) is constant with respect to \(\lambda\).
+The scaled Newton update is
+
+\[\lambda'(\alpha) = \lambda - \alpha \partial r(\lambda)^{-1} r(\lambda)\]
+which requires evaluating \(\partial c\) on each iteration and solving a linear
+system in the residual Jacobian \(\partial r(\lambda)\).
+The step size \(\alpha \in [0, 1]\) is initialised at \(\alpha = 1\) with a
+backtracking line search performed by multiplying \(\alpha\) by 0.5 until
+\(|r(\lambda'(\alpha))| < |r(\lambda)\).
- Parameters:
-state (ChainState) – Post h2_flow update state to project.
state_prev (ChainState) – Previous state in co-tangent bundle before h2_flow update which
-defines the co-tangent space to perform projection in.
time_step (float) – Integrator time step used in h2_flow update.
system (System) – Hamiltonian system defining h2_flow and constr functions used to
-define constraint equation to solve.
state (ChainState) – Post h2_flow update state to project.
state_prev (ChainState) – Previous state in co-tangent bundle before h2_flow update
+which defines the co-tangent space to perform projection in.
time_step (float) – Integrator time step used in h2_flow update.
system (ConstrainedEuclideanMetricSystem) – Hamiltonian system defining h2_flow and constr functions
+used to define constraint equation to solve.
constraint_tol (float) – Convergence tolerance in constraint space. Iteration will
-continue until norm(constr(pos)) < constraint_tol where pos is the
-position at the current iteration.
norm(constr(pos)) < constraint_tol where pos
+is the position at the current iteration.
position_tol (float) – Convergence tolerance in position space. Iteration will continue
-until norm(delt_pos) < position_tol where delta_pos is the change in the
-position in the current iteration.
norm(delta_pos) < position_tol where delta_pos is the
+change in the position in the current iteration.
divergence_tol (float) – Divergence tolerance - solver aborts if
-norm(constr(pos)) > divergence_tol on any iteration where pos is the
-position at the current iteration and raises mici.errors.ConvergenceError.
norm(constr(pos)) > divergence_tol on any iteration where
+pos is the position at the current iteration and raises
+mici.errors.ConvergenceError.
max_iters (int) – Maximum number of iterations to perform before aborting and raising
-mici.errors.ConvergenceError.
mici.errors.ConvergenceError.
max_line_search_iters (int) – Maximum number of ‘inner’ line search iterations to
perform to try to find step size along search direction which decreases
residual norm.
- Returns:
Updated state object with position component satisfying constraint equation to
-within constraint_tol, i.e. norm(system.constr(state.pos)) < constraint_tol.
+within constraint_tol, i.e.
+norm(system.constr(state.pos)) < constraint_tol.
- Raises:
-ConvergenceError – If solver does not converge within max_iters
- iterations, diverges or encounters a ValueError during the iteration.
+ConvergenceError – If solver does not converge within
+ max_iters iterations, diverges or encounters a ValueError
+ during the iteration.
- Return type:
-
@@ -383,68 +402,84 @@
-
mici.solvers.solve_projection_onto_manifold_quasi_newton(state, state_prev, time_step, system, constraint_tol=1e-09, position_tol=1e-08, divergence_tol=10000000000.0, max_iters=50, norm=<function maximum_norm>)[source]
Solve constraint equation using symmetric quasi-Newton method.
-Only requires re-evaluating the constraint function system.constr within the
+
Only requires re-evaluating the constraint function system.constr within the
solver loop and no recomputation of matrix decompositions on each iteration.
Solves an equation of the form
-r(λ) = c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) = 0
-
-
-for the vector of Lagrange multipliers λ to project a point on to the manifold
-defined by the zero level set of a constraint function c (system.constr), with
-Φ₂,₁ the position component of a flow map for the h₂ Hamiltonian component for
-the system (i.e. system.h2_flow).
-The Jacobian of the residual function r is
-∂r(λ) = ∂c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) ∂₂Φ₂,₁ ∂c(q)ᵀ
-
-
-where ∂₂Φ₂,₁ is the Jacobian of the (linear) flow-map Φ₂,₁ with respect to the
-second (momentum argument), such that the full Newton update is
-λ_(α) = λ - ∂r(λ)⁻¹ r(λ)
-
-
-which requires evaluating ∂c (system.jacob_constr) on each iteration and solving
-a linear system in the residual Jacobian ∂r(λ).
+
+\[r(\lambda) = c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda)) = 0,\]
+for the vector of Lagrange multipliers \(\lambda\) to project a point on to the
+manifold defined by the zero level set of a constraint function \(c\), with
+\(\Phi_{2,1}\) the flow map for the \(h_2\) Hamiltonian component for the
+system restricted to the position component output.
+The Jacobian of the residual function \(r\) is
+
+\[\partial r(\lambda) =
+\partial c(\Phi_{2,1}(t)(q, p + \partial c(q)^T \lambda))
+\partial_2 (\Phi_{2,1}(t))(q, p + \partial c(q)^T \lambda)
+\partial c(q)^T\]
+where \(\partial_2 (\Phi_{2,1}(t))\) is the Jacobian of the (position
+restricted) flow-map \(\Phi_{2,1}\) with respect to its second (momentum)
+argument. We assume here that \(\Phi_{2,1}(t)\) is linear in its second
+(momentum) argument, such that \(\partial_2 (\Phi_{2,1}(t))(q, p + \partial
+c(q)^T \lambda)\) is constant with respect to \(\lambda\), and henceforth use the
+shorthand \(\partial_2 (\Phi_{2,1}(t))\) to refer to this matrix.
+The full Newton update is
+
+\[\lambda' = \lambda - \partial r(\lambda)^{-1} r(\lambda)\]
+which requires evaluating \(\partial c\) on each iteration and solving a linear
+system in the residual Jacobian \(\partial r(\lambda)\).
The symmetric quasi-Newton iteration instead uses the approximation
-∂c(Φ₂,₁(q, p + ∂c(q)ᵀλ)) ∂₂Φ₂,₁ ∂c(q)ᵀ ≈ ∂c(q) ∂₂Φ₂,₁ ∂c(q)ᵀ
-
-
+
+\[\partial c(\Phi_{2,1}(t)(q, p + \partial c(q)^T\lambda))
+\partial_2 (\Phi_{2,1}(t))
+\partial c(q)^T
+\approx
+\partial c(q)
+\partial_2 (\Phi_{2,1}(t))
+\partial c(q)^T\]
with the corresponding update
-λ = λ - (∂c(q) ∂₂Φ₂,₁ ∂c(q)ᵀ)⁻¹ r(λ)
-
-
-allowing a previously computed decomposition of the matrix ∂c(q) ∂₂Φ₂,₁ ∂c(q)ᵀ to
-be used to solve the linear system in each iteration with no requirement to evaluate
-∂c (system.jacob_constr) on each iteration.
+
+\[\lambda' =
+\lambda
+- (\partial c(q) \partial_2 (\Phi_{2,1}(t)) \partial c(q)^T)^{-1} r(\lambda)\]
+allowing a previously computed decomposition of the matrix
+
+\[\partial c(q) \partial_2 (\Phi_{2,1}(t)) \partial c(q)^T,\]
+to be used to solve the linear system in each iteration with no requirement to
+evaluate \(\partial c\) (system.jacob_constr) on each iteration.
- Parameters:
-state (ChainState) – Post h2_flow update state to project.
state_prev (ChainState) – Previous state in co-tangent bundle before h2_flow update which
-defines the co-tangent space to perform projection in.
time_step (float) – Integrator time step used in h2_flow update.
system (System) – Hamiltonian system defining h2_flow and constr functions used to
-define constraint equation to solve.
state (ChainState) – Post h2_flow update state to project.
state_prev (ChainState) – Previous state in co-tangent bundle before h2_flow update
+which defines the co-tangent space to perform projection in.
time_step (float) – Integrator time step used in h2_flow update.
system (ConstrainedEuclideanMetricSystem) – Hamiltonian system defining h2_flow and constr functions
+used to define constraint equation to solve.
constraint_tol (float) – Convergence tolerance in constraint space. Iteration will
-continue until norm(constr(pos)) < constraint_tol where pos is the
-position at the current iteration.
norm(constr(pos)) < constraint_tol where pos
+is the position at the current iteration.
position_tol (float) – Convergence tolerance in position space. Iteration will continue
-until norm(delt_pos) < position_tol where delta_pos is the change in the
-position in the current iteration.
norm(delta_pos) < position_tol where delta_pos is the
+change in the position in the current iteration.
divergence_tol (float) – Divergence tolerance - solver aborts if
-norm(constr(pos)) > divergence_tol on any iteration where pos is the
-position at the current iteration and raises mici.errors.ConvergenceError.
norm(constr(pos)) > divergence_tol on any iteration where
+pos is the position at the current iteration and raises
+mici.errors.ConvergenceError.
max_iters (int) – Maximum number of iterations to perform before aborting and raising
-mici.errors.ConvergenceError.
mici.errors.ConvergenceError.
norm (ScalarFunction) – Norm to use to test for convergence.
- Returns:
Updated state object with position component satisfying constraint equation to
-within constraint_tol, i.e. norm(system.constr(state.pos)) < constraint_tol.
+within constraint_tol, i.e.
+norm(system.constr(state.pos)) < constraint_tol.
- Raises:
-ConvergenceError – If solver does not converge within max_iters
- iterations, diverges or encounters a ValueError during the iteration.
+ConvergenceError – If solver does not converge within
+ max_iters iterations, diverges or encounters a ValueError
+ during the iteration.
- Return type:
-
diff --git a/mici.systems.html b/mici.systems.html
index 28061fc..400d4be 100644
--- a/mici.systems.html
+++ b/mici.systems.html
@@ -63,6 +63,7 @@
- mici.systems module
CholeskyFactoredRiemannianMetricSystemConstrainedEuclideanMetricSystemConstrainedTractableFlowSystemDenseConstrainedEuclideanMetricSystemDenseRiemannianMetricSystemDiagonalRiemannianMetricSystem
-
class mici.systems.ConstrainedEuclideanMetricSystem(neg_log_dens, constr, metric=None, dens_wrt_hausdorff=True, grad_neg_log_dens=None, jacob_constr=None)[source]
-Bases: EuclideanMetricSystem
+Bases: ConstrainedTractableFlowSystem, EuclideanMetricSystem
Base class for Euclidean Hamiltonian systems subject to constraints.
The (constrained) position space is assumed to be a differentiable manifold embedded
with a \(Q\)-dimensional ambient Euclidean space. The \(Q-C\) dimensional
@@ -457,7 +458,7 @@
\[h_2(q, p) = \frac{1}{2} p^T M^{-1} p.\]
The time-derivative of the constraint equation implies a further set of constraints
-on the momentum \(q\) with \(\partial c(q) M^{-1} p = 0\) at all time
+on the momentum \(p\) with \(\partial c(q) M^{-1} p = 0\) at all time
points, corresponding to the momentum (velocity) being in the co-tangent space
(tangent space) to the manifold.
The target distribution is either assumed to be directly specified with unnormalized
@@ -633,19 +634,22 @@
-
-dh2_flow_dmom(dt)
-Derivatives of h2_flow flow map with respect to momentum argument.
+dh2_flow_dmom(state, dt)[source]
+Derivatives of h2_flow() flow map with respect to momentum argument.
- Parameters:
-dt (ScalarLike) – Time interval flow simulated for.
+
+state (ChainState) – State to evaluate derivatives of flow map at.
dt (ScalarLike) – Time interval flow simulated for.
- Returns:
-Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix representing
-derivative (Jacobian) of position output of h2_flow() with respect
-to the value of the momentum component of the initial input state and
-dmom_dmom a matrix representing derivative (Jacobian) of momentum
-output of h2_flow() with respect to the value of the momentum
-component of the initial input state.
+Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix
+representing derivative (Jacobian) of position output of h2_flow()
+with respect to the value of the momentum component of the initial input
+state and dmom_dmom a matrix representing derivative (Jacobian) of
+momentum output of h2_flow() with respect to the value of the
+momentum component of the initial input state.
- Return type:
-
@@ -863,7 +867,7 @@
-
-abstract jacob_constr_inner_product(jacob_constr_1, inner_product_matrix, jacob_constr_2=None)[source]
+abstract jacob_constr_inner_product(jacob_constr_1, inner_product_matrix, jacob_constr_2=None)
Compute inner product of rows of constraint Jacobian matrices.
Computes jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.T
potentially exploiting any structure / sparsity in jacob_constr_1,
@@ -943,7 +947,393 @@
-
-sample_momentum(state, rng)[source]
+sample_momentum(state, rng)
+Sample a momentum from its conditional distribution given a position.
+
+- Parameters:
+
+state (ChainState) – State defining position to condition on.
rng (Generator) –
+- Returns:
+Sampled momentum.
+- Return type:
+ArrayLike
+
+
+
+
+
+
+-
+class mici.systems.ConstrainedTractableFlowSystem(neg_log_dens, grad_neg_log_dens=None)[source]
+Bases: TractableFlowSystem
+Base class for Hamiltonian systems subject to constraints with tractable flows.
+The (constrained) position space is assumed to be a differentiable manifold embedded
+with a \(Q\)-dimensional ambient Euclidean space. The \(Q-C\) dimensional
+manifold \(\mathcal{M}\) is implicitly defined by an equation
+
+\[\mathcal{M} = \lbrace q \in \mathbb{R}^Q : c(q) = 0 \rbrace`\]
+with \(c: \mathbb{R}^Q \to \mathbb{R}^C\) the differentiable and surjective
+vector-valued constraint function.
+The Hamiltonian function \(h\) is assumed to have the general form
+
+\[h(q, p) = h_1(q) + h_2(q, p)\]
+where \(q\) and \(p\) are the position and momentum variables respectively,
+and \(h_1\) and \(h_2\) Hamiltonian component functions. The exact
+unconstrained Hamiltonian flows for both the \(h_1\) and \(h_2\)
+components, respectively \(\Phi_1\) and \(\Phi_2\) are assumed to be
+tractable for subclasses of this class.
+The constrained Hamiltonian dynamics are described by the system of differential
+algebraic equations
+
+\[\dot{q} = \nabla_2 h(q, p), \quad
+\dot{p} = -\nabla_1 h(q, p) - \partial c(q)^T\lambda, \quad
+c(q) = 0.\]
+where math:lambda is a set of Lagrange multipliers of dimension equal to number
+of constraints, \(C\), and which are implicitly defined by the condition that
+the constraint equation \(c(q) = 0\) applies at all times.
+The dynamics implicitly define a set of constraints on the momentum variables,
+differentiating the constraint equation with respect to time giving that
+
+\[\partial c(q) \nabla_2 h(q, p) = \partial c(q) \nabla_2 h_2(q, p) = 0.\]
+The set of momentum variables satisfying the above for given position variables is
+termed the cotangent space of the manifold (at a position). Here we assume that the
+operation of projecting a momentum vector onto the cotangent space at a given
+position is tractable to compute.
+
+- Parameters:
+
+neg_log_dens (ScalarFunction) – Function which given a position array returns the negative
+logarithm of an unnormalized probability density on the position space
+with respect to a reference measure, with the corresponding
+distribution on the position space being the target distribution it is
+wished to draw approximate samples from.
grad_neg_log_dens (Optional[GradientFunction]) – Function which given a position array returns the
+derivative of neg_log_dens with respect to the position array
+argument. Optionally the function may instead return a 2-tuple of values
+with the first being the array corresponding to the derivative and the
+second being the value of the neg_log_dens evaluated at the passed
+position array. If None is passed (the default) an automatic
+differentiation fallback will be used to attempt to construct the
+derivative of neg_log_dens automatically.
+
+
+-
+abstract constr(state)[source]
+Constraint function at the current position.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of constr(state.pos) as 1D array.
+- Return type:
+ArrayLike
+
+
+
+
+-
+dh1_dpos(state)
+Derivative of h1 Hamiltonian component with respect to position.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of computed h1 derivative.
+- Return type:
+ScalarLike
+
+
+
+
+-
+abstract dh2_dmom(state)
+Derivative of h2 Hamiltonian component with respect to momentum.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of h2(state) derivative with respect to state.mom.
+- Return type:
+ArrayLike
+
+
+
+
+-
+abstract dh2_dpos(state)
+Derivative of h2 Hamiltonian component with respect to position.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of h2(state) derivative with respect to state.pos.
+- Return type:
+ArrayLike
+
+
+
+
+-
+abstract dh2_flow_dmom(state, dt)[source]
+Derivatives of h2_flow() flow map with respect to momentum argument.
+
+- Parameters:
+
+state (ChainState) – State to evaluate derivatives of flow map at.
dt (ScalarLike) – Time interval flow simulated for.
+- Returns:
+Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix
+representing derivative (Jacobian) of position output of h2_flow()
+with respect to the value of the momentum component of the initial input
+state and dmom_dmom a matrix representing derivative (Jacobian) of
+momentum output of h2_flow() with respect to the value of the
+momentum component of the initial input state.
+- Return type:
+-
+
+
+
+
+
+-
+dh_dmom(state)
+Derivative of Hamiltonian with respect to momentum.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of h(state) derivative with respect to state.mom.
+- Return type:
+ArrayLike
+
+
+
+
+-
+dh_dpos(state)
+Derivative of Hamiltonian with respect to position.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of h(state) derivative with respect to state.pos.
+- Return type:
+ArrayLike
+
+
+
+
+-
+grad_neg_log_dens(state)
+Derivative of negative log density with respect to position.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of neg_log_dens(state) derivative with respect to state.pos.
+- Return type:
+ArrayLike
+
+
+
+
+-
+h(state)
+Hamiltonian function for system.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of Hamiltonian.
+- Return type:
+ScalarLike
+
+
+
+
+-
+h1(state)
+Hamiltonian component depending only on position.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of h1 Hamiltonian component.
+- Return type:
+ScalarLike
+
+
+
+
+-
+h1_flow(state, dt)
+Apply exact flow map corresponding to h1 Hamiltonian component.
+state argument is modified in place.
+
+- Parameters:
+
+state (ChainState) – State to start flow at.
dt (ScalarLike) – Time interval to simulate flow for.
+
+
+
+
+-
+abstract h2(state)
+Hamiltonian component depending on momentum and optionally position.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of h2 Hamiltonian component.
+- Return type:
+ScalarLike
+
+
+
+
+-
+abstract h2_flow(state, dt)
+Apply exact flow map corresponding to h2 Hamiltonian component.
+state argument is modified in place.
+
+- Parameters:
+
+state (ChainState) – State to start flow at.
dt (ScalarLike) – Time interval to simulate flow for.
+
+
+
+
+-
+abstract jacob_constr(state)[source]
+Jacobian of constraint function at the current position.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of Jacobian of constr(state.pos) as 2D array.
+- Return type:
+ArrayLike
+
+
+
+
+-
+abstract jacob_constr_inner_product(jacob_constr_1, inner_product_matrix, jacob_constr_2=None)[source]
+Compute inner product of rows of constraint Jacobian matrices.
+Computes jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.T
+potentially exploiting any structure / sparsity in jacob_constr_1,
+jacob_constr_2 and inner_product_matrix.
+
+- Parameters:
+
+jacob_constr_1 (MatrixLike) – First constraint Jacobian in product.
inner_product_matrix (PositiveDefiniteMatrix) – Positive-definite matrix defining inner-product
+between rows of two constraint Jacobians.
jacob_constr_2 (Optional[MatrixLike]) – Second constraint Jacobian in product. Defaults to
+jacob_constr_1 if set to None.
+- Return type:
+MatrixLike
+
+
+- Returns
Object corresponding to computed inner products of the constraint Jacobian
+rows.
+
+
+
+
+-
+neg_log_dens(state)
+Negative logarithm of unnormalized density of target distribution.
+
+- Parameters:
+state (ChainState) – State to compute value at.
+- Returns:
+Value of computed negative log density.
+- Return type:
+ScalarLike
+
+
+
+
+-
+abstract project_onto_cotangent_space(mom, state)[source]
+Project a momentum on to the co-tangent space at a position.
+
+- Parameters:
+
+mom (ArrayLike) – Momentum (co-)vector as 1D array to project on to co-tangent space.
state (ChainState) – State definining position on the manifold to project in to the
+co-tangent space of.
+- Returns:
+Projected momentum in the co-tangent space at state.pos.
+- Return type:
+ArrayLike
+
+
+
+
+-
+sample_momentum(state, rng)[source]
Sample a momentum from its conditional distribution given a position.
- Parameters:
@@ -1131,19 +1521,22 @@
-
-dh2_flow_dmom(dt)
-Derivatives of h2_flow flow map with respect to momentum argument.
+dh2_flow_dmom(state, dt)
+Derivatives of h2_flow() flow map with respect to momentum argument.
- Parameters:
-dt (ScalarLike) – Time interval flow simulated for.
+
+state (ChainState) – State to evaluate derivatives of flow map at.
dt (ScalarLike) – Time interval flow simulated for.
- Returns:
-Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix representing
-derivative (Jacobian) of position output of h2_flow() with respect
-to the value of the momentum component of the initial input state and
-dmom_dmom a matrix representing derivative (Jacobian) of momentum
-output of h2_flow() with respect to the value of the momentum
-component of the initial input state.
+Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix
+representing derivative (Jacobian) of position output of h2_flow()
+with respect to the value of the momentum component of the initial input
+state and dmom_dmom a matrix representing derivative (Jacobian) of
+momentum output of h2_flow() with respect to the value of the
+momentum component of the initial input state.
- Return type:
-
@@ -2223,28 +2616,6 @@
-
--
-dh2_flow_dmom(dt)[source]
-Derivatives of h2_flow flow map with respect to momentum argument.
-
-- Parameters:
-dt (ScalarLike) – Time interval flow simulated for.
-- Returns:
-Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix representing
-derivative (Jacobian) of position output of h2_flow() with respect
-to the value of the momentum component of the initial input state and
-dmom_dmom a matrix representing derivative (Jacobian) of momentum
-output of h2_flow() with respect to the value of the momentum
-component of the initial input state.
-- Return type:
--
-
-
-
-
-
dh_dmom(state)
@@ -2420,9 +2791,9 @@
-
class mici.systems.GaussianDenseConstrainedEuclideanMetricSystem(neg_log_dens, constr, metric=None, grad_neg_log_dens=None, jacob_constr=None, mhp_constr=None)[source]
Bases: GaussianEuclideanMetricSystem, DenseConstrainedEuclideanMetricSystem
-Gaussian Euclidean Hamiltonian system st. a dense set of constraints.
-See ConstrainedEuclideanMetricSystem for more details about constrained
-systems and GaussianEuclideanMetricSystem for Gaussian Euclidean metric
+
Gaussian Euclidean Hamiltonian system subject to a dense set of constraints.
+See ConstrainedEuclideanMetricSystem for more details about constrained
+systems and GaussianEuclideanMetricSystem for Gaussian Euclidean metric
systems.
- Parameters:
@@ -2576,19 +2947,22 @@
-
-dh2_flow_dmom(dt)
-Derivatives of h2_flow flow map with respect to momentum argument.
+dh2_flow_dmom(state, dt)[source]
+Derivatives of h2_flow() flow map with respect to momentum argument.
- Parameters:
-dt (ScalarLike) – Time interval flow simulated for.
+
+state (ChainState) – State to evaluate derivatives of flow map at.
dt (ScalarLike) – Time interval flow simulated for.
- Returns:
-Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix representing
-derivative (Jacobian) of position output of h2_flow() with respect
-to the value of the momentum component of the initial input state and
-dmom_dmom a matrix representing derivative (Jacobian) of momentum
-output of h2_flow() with respect to the value of the momentum
-component of the initial input state.
+Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix
+representing derivative (Jacobian) of position output of h2_flow()
+with respect to the value of the momentum component of the initial input
+state and dmom_dmom a matrix representing derivative (Jacobian) of
+momentum output of h2_flow() with respect to the value of the
+momentum component of the initial input state.
- Return type:
-
@@ -3029,28 +3403,6 @@
-
--
-dh2_flow_dmom(dt)[source]
-Derivatives of h2_flow flow map with respect to momentum argument.
-
-- Parameters:
-dt (ScalarLike) – Time interval flow simulated for.
-- Returns:
-Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix representing
-derivative (Jacobian) of position output of h2_flow() with respect
-to the value of the momentum component of the initial input state and
-dmom_dmom a matrix representing derivative (Jacobian) of momentum
-output of h2_flow() with respect to the value of the momentum
-component of the initial input state.
-- Return type:
--
-
-
-
-
-
dh_dmom(state)
@@ -4656,28 +5008,6 @@
-
--
-abstract dh2_flow_dmom(dt)[source]
-Derivatives of h2_flow flow map with respect to momentum argument.
-
-- Parameters:
-dt (ScalarLike) – Time interval flow simulated for.
-- Returns:
-Tuple (dpos_dmom, dmom_dmom) with dpos_dmom a matrix representing
-derivative (Jacobian) of position output of h2_flow() with respect
-to the value of the momentum component of the initial input state and
-dmom_dmom a matrix representing derivative (Jacobian) of momentum
-output of h2_flow() with respect to the value of the momentum
-component of the initial input state.
-- Return type:
--
-
-
-
-
-
dh_dmom(state)
diff --git a/mici.transitions.html b/mici.transitions.html
index 6b63491..ee644b4 100644
--- a/mici.transitions.html
+++ b/mici.transitions.html
@@ -273,7 +273,7 @@
-
-property state_variables: Set[str]
+property state_variables: set[str]
A set of names of state variables accessed by this transition.
@@ -391,7 +391,7 @@
-
-property state_variables: Set[str]
+property state_variables: set[str]
A set of names of state variables accessed by this transition.
@@ -458,7 +458,7 @@
-
-property state_variables: Set[str]
+property state_variables: set[str]
A set of names of state variables accessed by this transition.
@@ -532,7 +532,7 @@
-
-property state_variables: Set[str]
+property state_variables: set[str]
A set of names of state variables accessed by this transition.
@@ -601,7 +601,7 @@
-
-property state_variables: Set[str]
+property state_variables: set[str]
A set of names of state variables accessed by this transition.
@@ -757,7 +757,7 @@
@@ -854,7 +854,7 @@
diff --git a/objects.inv b/objects.inv
index 1ee99b8..044d406 100644
Binary files a/objects.inv and b/objects.inv differ
diff --git a/searchindex.js b/searchindex.js
index 09ad498..d130551 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
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\ No newline at end of file
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"module-mici.errors"]], "with_traceback() (mici.errors.adaptationerror method)": [[5, "mici.errors.AdaptationError.with_traceback"]], "with_traceback() (mici.errors.convergenceerror method)": [[5, "mici.errors.ConvergenceError.with_traceback"]], "with_traceback() (mici.errors.error method)": [[5, "mici.errors.Error.with_traceback"]], "with_traceback() (mici.errors.hamiltoniandivergenceerror method)": [[5, "mici.errors.HamiltonianDivergenceError.with_traceback"]], "with_traceback() (mici.errors.integratorerror method)": [[5, "mici.errors.IntegratorError.with_traceback"]], "with_traceback() (mici.errors.linalgerror method)": [[5, "mici.errors.LinAlgError.with_traceback"]], "with_traceback() (mici.errors.nonreversiblesteperror method)": [[5, "mici.errors.NonReversibleStepError.with_traceback"]], "with_traceback() (mici.errors.readonlystateerror method)": [[5, "mici.errors.ReadOnlyStateError.with_traceback"]], "bcssfourstageintegrator (class in mici.integrators)": [[6, 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mici.matrices)": [[8, "mici.matrices.DensePositiveDefiniteMatrix"]], "densepositivedefiniteproductmatrix (class in mici.matrices)": [[8, "mici.matrices.DensePositiveDefiniteProductMatrix"]], "denserectangularmatrix (class in mici.matrices)": [[8, "mici.matrices.DenseRectangularMatrix"]], "densesquarematrix (class in mici.matrices)": [[8, "mici.matrices.DenseSquareMatrix"]], "densesymmetricmatrix (class in mici.matrices)": [[8, "mici.matrices.DenseSymmetricMatrix"]], "diagonalmatrix (class in mici.matrices)": [[8, "mici.matrices.DiagonalMatrix"]], "differentiablematrix (class in mici.matrices)": [[8, "mici.matrices.DifferentiableMatrix"]], "eigendecomposedpositivedefinitematrix (class in mici.matrices)": [[8, "mici.matrices.EigendecomposedPositiveDefiniteMatrix"]], "eigendecomposedsymmetricmatrix (class in mici.matrices)": [[8, "mici.matrices.EigendecomposedSymmetricMatrix"]], "explicitarraymatrix (class in mici.matrices)": [[8, "mici.matrices.ExplicitArrayMatrix"]], "identitymatrix (class in mici.matrices)": [[8, "mici.matrices.IdentityMatrix"]], "implicitarraymatrix (class in mici.matrices)": [[8, "mici.matrices.ImplicitArrayMatrix"]], "inverselufactoredsquarematrix (class in mici.matrices)": [[8, "mici.matrices.InverseLUFactoredSquareMatrix"]], "inversetriangularmatrix (class in mici.matrices)": [[8, "mici.matrices.InverseTriangularMatrix"]], "invertiblematrix (class in mici.matrices)": [[8, "mici.matrices.InvertibleMatrix"]], "invertiblematrixproduct (class in mici.matrices)": [[8, "mici.matrices.InvertibleMatrixProduct"]], "matrix (class in mici.matrices)": [[8, "mici.matrices.Matrix"]], "matrixproduct (class in mici.matrices)": [[8, "mici.matrices.MatrixProduct"]], "orthogonalmatrix (class in mici.matrices)": [[8, "mici.matrices.OrthogonalMatrix"]], "positivedefiniteblockdiagonalmatrix (class in mici.matrices)": [[8, "mici.matrices.PositiveDefiniteBlockDiagonalMatrix"]], "positivedefinitelowrankupdatematrix (class in mici.matrices)": [[8, "mici.matrices.PositiveDefiniteLowRankUpdateMatrix"]], "positivedefinitematrix (class in mici.matrices)": [[8, "mici.matrices.PositiveDefiniteMatrix"]], "positivediagonalmatrix (class in mici.matrices)": [[8, "mici.matrices.PositiveDiagonalMatrix"]], "positivescaledidentitymatrix (class in mici.matrices)": [[8, "mici.matrices.PositiveScaledIdentityMatrix"]], "scaledidentitymatrix (class in mici.matrices)": [[8, "mici.matrices.ScaledIdentityMatrix"]], "scaledorthogonalmatrix (class in mici.matrices)": [[8, "mici.matrices.ScaledOrthogonalMatrix"]], "softabsregularizedpositivedefinitematrix (class in mici.matrices)": [[8, "mici.matrices.SoftAbsRegularizedPositiveDefiniteMatrix"]], "squareblockdiagonalmatrix (class in mici.matrices)": [[8, "mici.matrices.SquareBlockDiagonalMatrix"]], "squarelowrankupdatematrix (class in mici.matrices)": [[8, "mici.matrices.SquareLowRankUpdateMatrix"]], "squarematrix (class in mici.matrices)": [[8, "mici.matrices.SquareMatrix"]], "squarematrixproduct (class in mici.matrices)": [[8, "mici.matrices.SquareMatrixProduct"]], "symmetricblockdiagonalmatrix (class in mici.matrices)": [[8, "mici.matrices.SymmetricBlockDiagonalMatrix"]], "symmetriclowrankupdatematrix (class in mici.matrices)": [[8, "mici.matrices.SymmetricLowRankUpdateMatrix"]], "symmetricmatrix (class in mici.matrices)": [[8, "mici.matrices.SymmetricMatrix"]], "t (mici.matrices.blockcolumnmatrix property)": [[8, "mici.matrices.BlockColumnMatrix.T"]], "t (mici.matrices.blockmatrix property)": [[8, "mici.matrices.BlockMatrix.T"]], "t (mici.matrices.blockrowmatrix property)": [[8, "mici.matrices.BlockRowMatrix.T"]], "t (mici.matrices.densedefinitematrix property)": [[8, "mici.matrices.DenseDefiniteMatrix.T"]], "t (mici.matrices.densepositivedefinitematrix property)": [[8, "mici.matrices.DensePositiveDefiniteMatrix.T"]], "t (mici.matrices.densepositivedefiniteproductmatrix property)": [[8, "mici.matrices.DensePositiveDefiniteProductMatrix.T"]], "t (mici.matrices.denserectangularmatrix property)": [[8, "mici.matrices.DenseRectangularMatrix.T"]], "t (mici.matrices.densesquarematrix property)": [[8, "mici.matrices.DenseSquareMatrix.T"]], "t (mici.matrices.densesymmetricmatrix property)": [[8, "mici.matrices.DenseSymmetricMatrix.T"]], "t (mici.matrices.diagonalmatrix property)": [[8, "mici.matrices.DiagonalMatrix.T"]], "t (mici.matrices.differentiablematrix property)": [[8, "mici.matrices.DifferentiableMatrix.T"]], "t (mici.matrices.eigendecomposedpositivedefinitematrix property)": [[8, "mici.matrices.EigendecomposedPositiveDefiniteMatrix.T"]], "t (mici.matrices.eigendecomposedsymmetricmatrix property)": [[8, "mici.matrices.EigendecomposedSymmetricMatrix.T"]], "t (mici.matrices.explicitarraymatrix property)": [[8, "mici.matrices.ExplicitArrayMatrix.T"]], "t (mici.matrices.identitymatrix property)": [[8, "mici.matrices.IdentityMatrix.T"]], "t (mici.matrices.implicitarraymatrix property)": [[8, "mici.matrices.ImplicitArrayMatrix.T"]], "t (mici.matrices.inverselufactoredsquarematrix property)": [[8, "mici.matrices.InverseLUFactoredSquareMatrix.T"]], "t (mici.matrices.inversetriangularmatrix property)": [[8, "mici.matrices.InverseTriangularMatrix.T"]], "t (mici.matrices.invertiblematrix property)": [[8, "mici.matrices.InvertibleMatrix.T"]], "t (mici.matrices.invertiblematrixproduct property)": [[8, "mici.matrices.InvertibleMatrixProduct.T"]], "t (mici.matrices.matrix property)": [[8, "mici.matrices.Matrix.T"]], "t (mici.matrices.matrixproduct property)": [[8, "mici.matrices.MatrixProduct.T"]], "t (mici.matrices.orthogonalmatrix property)": [[8, "mici.matrices.OrthogonalMatrix.T"]], "t (mici.matrices.positivedefiniteblockdiagonalmatrix property)": [[8, "mici.matrices.PositiveDefiniteBlockDiagonalMatrix.T"]], "t (mici.matrices.positivedefinitelowrankupdatematrix property)": [[8, "mici.matrices.PositiveDefiniteLowRankUpdateMatrix.T"]], "t (mici.matrices.positivedefinitematrix property)": [[8, "mici.matrices.PositiveDefiniteMatrix.T"]], "t (mici.matrices.positivediagonalmatrix property)": [[8, "mici.matrices.PositiveDiagonalMatrix.T"]], "t (mici.matrices.positivescaledidentitymatrix property)": [[8, "mici.matrices.PositiveScaledIdentityMatrix.T"]], "t (mici.matrices.scaledidentitymatrix property)": [[8, "mici.matrices.ScaledIdentityMatrix.T"]], "t (mici.matrices.scaledorthogonalmatrix property)": [[8, "mici.matrices.ScaledOrthogonalMatrix.T"]], "t (mici.matrices.softabsregularizedpositivedefinitematrix property)": [[8, "mici.matrices.SoftAbsRegularizedPositiveDefiniteMatrix.T"]], "t (mici.matrices.squareblockdiagonalmatrix property)": [[8, "mici.matrices.SquareBlockDiagonalMatrix.T"]], "t (mici.matrices.squarelowrankupdatematrix property)": [[8, "mici.matrices.SquareLowRankUpdateMatrix.T"]], "t (mici.matrices.squarematrix property)": [[8, "mici.matrices.SquareMatrix.T"]], "t (mici.matrices.squarematrixproduct property)": [[8, "mici.matrices.SquareMatrixProduct.T"]], "t (mici.matrices.symmetricblockdiagonalmatrix property)": [[8, "mici.matrices.SymmetricBlockDiagonalMatrix.T"]], "t (mici.matrices.symmetriclowrankupdatematrix property)": [[8, "mici.matrices.SymmetricLowRankUpdateMatrix.T"]], "t (mici.matrices.symmetricmatrix property)": [[8, "mici.matrices.SymmetricMatrix.T"]], "t (mici.matrices.triangularfactoreddefinitematrix property)": [[8, "mici.matrices.TriangularFactoredDefiniteMatrix.T"]], "t (mici.matrices.triangularfactoredpositivedefinitematrix property)": [[8, "mici.matrices.TriangularFactoredPositiveDefiniteMatrix.T"]], "t (mici.matrices.triangularmatrix property)": [[8, "mici.matrices.TriangularMatrix.T"]], "triangularfactoreddefinitematrix (class in mici.matrices)": [[8, "mici.matrices.TriangularFactoredDefiniteMatrix"]], "triangularfactoredpositivedefinitematrix (class in mici.matrices)": [[8, "mici.matrices.TriangularFactoredPositiveDefiniteMatrix"]], "triangularmatrix (class in mici.matrices)": [[8, "mici.matrices.TriangularMatrix"]], "array (mici.matrices.blockcolumnmatrix property)": [[8, "mici.matrices.BlockColumnMatrix.array"]], "array (mici.matrices.blockmatrix property)": [[8, "mici.matrices.BlockMatrix.array"]], "array (mici.matrices.blockrowmatrix property)": [[8, "mici.matrices.BlockRowMatrix.array"]], "array (mici.matrices.densedefinitematrix property)": [[8, "mici.matrices.DenseDefiniteMatrix.array"]], "array (mici.matrices.densepositivedefinitematrix property)": [[8, "mici.matrices.DensePositiveDefiniteMatrix.array"]], "array (mici.matrices.densepositivedefiniteproductmatrix property)": [[8, "mici.matrices.DensePositiveDefiniteProductMatrix.array"]], "array (mici.matrices.denserectangularmatrix property)": [[8, "mici.matrices.DenseRectangularMatrix.array"]], "array (mici.matrices.densesquarematrix property)": [[8, "mici.matrices.DenseSquareMatrix.array"]], "array (mici.matrices.densesymmetricmatrix property)": [[8, "mici.matrices.DenseSymmetricMatrix.array"]], "array (mici.matrices.diagonalmatrix property)": [[8, "mici.matrices.DiagonalMatrix.array"]], "array (mici.matrices.differentiablematrix property)": [[8, "mici.matrices.DifferentiableMatrix.array"]], "array (mici.matrices.eigendecomposedpositivedefinitematrix property)": [[8, "mici.matrices.EigendecomposedPositiveDefiniteMatrix.array"]], "array (mici.matrices.eigendecomposedsymmetricmatrix property)": [[8, "mici.matrices.EigendecomposedSymmetricMatrix.array"]], "array (mici.matrices.explicitarraymatrix property)": [[8, "mici.matrices.ExplicitArrayMatrix.array"]], "array (mici.matrices.identitymatrix property)": [[8, "mici.matrices.IdentityMatrix.array"]], "array (mici.matrices.implicitarraymatrix property)": [[8, "mici.matrices.ImplicitArrayMatrix.array"]], "array (mici.matrices.inverselufactoredsquarematrix property)": [[8, "mici.matrices.InverseLUFactoredSquareMatrix.array"]], "array (mici.matrices.inversetriangularmatrix property)": [[8, "mici.matrices.InverseTriangularMatrix.array"]], "array (mici.matrices.invertiblematrix property)": [[8, "mici.matrices.InvertibleMatrix.array"]], "array (mici.matrices.invertiblematrixproduct property)": [[8, "mici.matrices.InvertibleMatrixProduct.array"]], "array (mici.matrices.matrix property)": [[8, "mici.matrices.Matrix.array"]], "array (mici.matrices.matrixproduct property)": [[8, "mici.matrices.MatrixProduct.array"]], "array (mici.matrices.orthogonalmatrix property)": [[8, "mici.matrices.OrthogonalMatrix.array"]], "array (mici.matrices.positivedefiniteblockdiagonalmatrix property)": [[8, "mici.matrices.PositiveDefiniteBlockDiagonalMatrix.array"]], "array (mici.matrices.positivedefinitelowrankupdatematrix property)": [[8, "mici.matrices.PositiveDefiniteLowRankUpdateMatrix.array"]], "array (mici.matrices.positivedefinitematrix property)": [[8, "mici.matrices.PositiveDefiniteMatrix.array"]], "array (mici.matrices.positivediagonalmatrix property)": [[8, "mici.matrices.PositiveDiagonalMatrix.array"]], "array (mici.matrices.positivescaledidentitymatrix property)": [[8, "mici.matrices.PositiveScaledIdentityMatrix.array"]], "array (mici.matrices.scaledidentitymatrix property)": [[8, "mici.matrices.ScaledIdentityMatrix.array"]], "array (mici.matrices.scaledorthogonalmatrix property)": [[8, "mici.matrices.ScaledOrthogonalMatrix.array"]], "array (mici.matrices.softabsregularizedpositivedefinitematrix property)": [[8, "mici.matrices.SoftAbsRegularizedPositiveDefiniteMatrix.array"]], "array (mici.matrices.squareblockdiagonalmatrix property)": [[8, "mici.matrices.SquareBlockDiagonalMatrix.array"]], "array (mici.matrices.squarelowrankupdatematrix property)": [[8, "mici.matrices.SquareLowRankUpdateMatrix.array"]], "array (mici.matrices.squarematrix property)": [[8, "mici.matrices.SquareMatrix.array"]], "array (mici.matrices.squarematrixproduct property)": [[8, "mici.matrices.SquareMatrixProduct.array"]], "array (mici.matrices.symmetricblockdiagonalmatrix property)": [[8, "mici.matrices.SymmetricBlockDiagonalMatrix.array"]], "array (mici.matrices.symmetriclowrankupdatematrix property)": [[8, "mici.matrices.SymmetricLowRankUpdateMatrix.array"]], "array (mici.matrices.symmetricmatrix property)": [[8, "mici.matrices.SymmetricMatrix.array"]], "array (mici.matrices.triangularfactoreddefinitematrix property)": [[8, "mici.matrices.TriangularFactoredDefiniteMatrix.array"]], "array (mici.matrices.triangularfactoredpositivedefinitematrix property)": [[8, "mici.matrices.TriangularFactoredPositiveDefiniteMatrix.array"]], "array (mici.matrices.triangularmatrix property)": [[8, "mici.matrices.TriangularMatrix.array"]], "blocks (mici.matrices.blockcolumnmatrix property)": [[8, "mici.matrices.BlockColumnMatrix.blocks"]], "blocks (mici.matrices.blockmatrix property)": [[8, "mici.matrices.BlockMatrix.blocks"]], "blocks (mici.matrices.blockrowmatrix property)": [[8, "mici.matrices.BlockRowMatrix.blocks"]], "blocks (mici.matrices.positivedefiniteblockdiagonalmatrix 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(mici.matrices.blockrowmatrix property)": [[8, "mici.matrices.BlockRowMatrix.diagonal"]], "diagonal (mici.matrices.densedefinitematrix property)": [[8, "mici.matrices.DenseDefiniteMatrix.diagonal"]], "diagonal (mici.matrices.densepositivedefinitematrix property)": [[8, "mici.matrices.DensePositiveDefiniteMatrix.diagonal"]], "diagonal (mici.matrices.densepositivedefiniteproductmatrix property)": [[8, "mici.matrices.DensePositiveDefiniteProductMatrix.diagonal"]], "diagonal (mici.matrices.denserectangularmatrix property)": [[8, "mici.matrices.DenseRectangularMatrix.diagonal"]], "diagonal (mici.matrices.densesquarematrix property)": [[8, "mici.matrices.DenseSquareMatrix.diagonal"]], "diagonal (mici.matrices.densesymmetricmatrix property)": [[8, "mici.matrices.DenseSymmetricMatrix.diagonal"]], "diagonal (mici.matrices.diagonalmatrix property)": [[8, "mici.matrices.DiagonalMatrix.diagonal"]], "diagonal (mici.matrices.differentiablematrix property)": [[8, 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\ No newline at end of file