Source code for mici.stagers
-"""Classes for controlling sampling of Markov chains split into stages."""
+"""Classes for staging sampling of Markov chains."""
from __future__ import annotations
@@ -78,26 +78,24 @@ Source code for mici.stagers
from typing import NamedTuple, TYPE_CHECKING
if TYPE_CHECKING:
- from typing import Callable, Iterable
- from numpy.typing import ArrayLike
+ from typing import Iterable, Optional
from mici.adapters import Adapter
- from mici.states import ChainState
from mici.types import TraceFunction
[docs]class ChainStage(NamedTuple):
"""Parameters of chain stage.
-
+
Parameters:
n_iter: Number of iterations in chain stage.
adapters: Dictionary of adapters to apply to each transition in chain stage.
trace_funcs: Functions defining chain variables to trace during chain stage.
- record_stats: Whether to record statistics and traces during chain stage.
+ record_stats: Whether to record statistics during chain stage.
"""
n_iter: int
adapters: dict[str, Iterable[Adapter]]
- trace_funcs: tuple[TraceFunction]
+ trace_funcs: Optional[tuple[TraceFunction]]
record_stats: bool
@@ -115,10 +113,10 @@ Source code for mici.stagers
) -> dict[str, ChainStage]:
"""Create dictionary specifying labels and parameters of chain sampling stages.
- Constructs a (ordered) dictionary with entries corresponding to the sequence of
+ Constructs an ordered dictionary with entries corresponding to the sequence of
sampling stages when running chains with one or more initial adaptation stages.
The keys of each entry are a string label for the sampling stage and the value a
- `ChainStage` named tuple specifying the parameters of the stage.
+ :py:class:`ChainStage` named tuple specifying the parameters of the stage.
Args:
n_warm_up_iter: Number of adaptive warm up iterations per chain. Depending
@@ -136,16 +134,19 @@ Source code for mici.stagers
stored. By default chains are only traced in the iterations of the final
non-adaptive stage - this behaviour can be altered using the
`trace_warm_up` argument.
- adapters: Dictionary of iterables of `Adapter` instances keyed by strings
- corresponding to the key of the transition in the sampler `transitions`
+ adapters: Dictionary of iterables of :py:class:`mici.adapters.Adapter`
+ instances keyed by strings corresponding to the key of the transition in
+ the sampler
+ :py:attr:`mici.samplers.MarkovChainMonteCarloMethod.transitions`
dictionary to apply the adapters to, to use to adaptatively set
parameters of the transitions during the adaptive stages of the chains.
Note that the adapter updates are applied in the order the adapters
appear in the iterables and so if multiple adapters change the same
parameter(s) the order will matter.
trace_warm_up: Whether to record chain traces and statistics during warm-up
- stage iterations (`True`) or only record traces and statistics in the
- iterations of the final non-adaptive stage (`False`, the default).
+ stage iterations (:code:`True`) or only record traces and statistics in
+ the iterations of the final non-adaptive stage (:code:`False`, the
+ default).
Returns:
Dictionary specifying chain stages, keyed by labels for stages.
@@ -169,17 +170,18 @@ Source code for mici.stagers
n_warm_up_iter: int,
n_main_iter: int,
adapters: dict[str, Iterable[Adapter]],
- trace_funcs: Iterable[Callable[[ChainState], dict[str, ArrayLike]]],
+ trace_funcs: Iterable[TraceFunction],
trace_warm_up: bool = False,
) -> dict[str, ChainStage]:
sampling_stages = dict()
+ trace_funcs = tuple(trace_funcs) if trace_funcs is not None else trace_funcs
# adaptive warm up stage
if n_warm_up_iter > 0:
warm_up_trace_funcs = trace_funcs if trace_warm_up else None
sampling_stages["Adaptive warm up"] = ChainStage(
n_iter=n_warm_up_iter,
adapters=adapters,
- trace_funcs=tuple(warm_up_trace_funcs),
+ trace_funcs=warm_up_trace_funcs,
record_stats=trace_warm_up,
)
# main non-adaptive stage
@@ -187,7 +189,7 @@ Source code for mici.stagers
sampling_stages["Main non-adaptive"] = ChainStage(
n_iter=n_main_iter,
adapters=None,
- trace_funcs=tuple(trace_funcs),
+ trace_funcs=trace_funcs,
record_stats=True,
)
return sampling_stages
@@ -203,7 +205,8 @@ Source code for mici.stagers
information. The adapters to be used in both the fast and slow adaptation stages
will be referred to as the *fast adapters* and the adapters to use in only the slow
adaptation stages the *slow adapters*. Each adapter self identifies if it is a fast
- adapter by whether the `is_fast` attribute is set to `True`.
+ adapter by whether the :py:attr:`mici.adapters.Adapter.is_fast` attribute is set to
+ :code:`True`.
The adaptive warm up iterations are split into three stages:
@@ -267,6 +270,7 @@ Source code for mici.stagers
trace_funcs: Iterable[TraceFunction],
trace_warm_up: bool = False,
) -> dict[str, ChainStage]:
+ trace_funcs = tuple(trace_funcs) if trace_funcs is not None else trace_funcs
fast_adapters = {
trans_key: [adapter for adapter in adapter_list if adapter.is_fast]
for trans_key, adapter_list in adapters.items()
@@ -294,7 +298,7 @@ Source code for mici.stagers
sampling_stages["Initial fast adaptive"] = ChainStage(
n_iter=n_init_fast_stage_iter,
adapters=fast_adapters,
- trace_funcs=tuple(warm_up_trace_funcs),
+ trace_funcs=warm_up_trace_funcs,
record_stats=record_stats,
)
# growing size slow adaptation windows
@@ -323,14 +327,14 @@ Source code for mici.stagers
] = ChainStage(
n_iter=n_iter,
adapters=adapters,
- trace_funcs=tuple(warm_up_trace_funcs),
+ trace_funcs=warm_up_trace_funcs,
record_stats=record_stats,
)
# final fast adaptation stage
sampling_stages["Final fast adaptive"] = ChainStage(
n_iter=n_final_fast_stage_iter,
adapters=fast_adapters,
- trace_funcs=tuple(warm_up_trace_funcs),
+ trace_funcs=warm_up_trace_funcs,
record_stats=record_stats,
)
# main non-adaptive stage
@@ -338,7 +342,7 @@ Source code for mici.stagers
sampling_stages["Main non-adaptive"] = ChainStage(
n_iter=n_main_iter,
adapters=None,
- trace_funcs=tuple(trace_funcs),
+ trace_funcs=trace_funcs,
record_stats=True,
)
return sampling_stages
diff --git a/_modules/mici/systems.html b/_modules/mici/systems.html
index b27219f..b6a3bc7 100644
--- a/_modules/mici/systems.html
+++ b/_modules/mici/systems.html
@@ -120,7 +120,7 @@ Source code for mici.systems
the corresponding exact Hamiltonian flow may or may not be simulable.
By default :math:`h_1` is assumed to correspond to the negative logarithm of an
- unnormalized density on the position variables with respect to the Lebesgue measure,
+ unnormalized density on the position variables 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.
"""
@@ -134,7 +134,7 @@ Source code for mici.systems
Args:
neg_log_dens: Function which given a position array returns the negative
logarithm of an unnormalized probability density on the position space
- with respect to the Lebesgue measure, with the corresponding
+ 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: Function which given a position array returns the
@@ -288,7 +288,55 @@ Source code for mici.systems
"""
-[docs]class EuclideanMetricSystem(System):
+[docs]class TractableFlowSystem(System):
+ r"""Base class for Hamiltonian systems with tractable component flows.
+
+ 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
+ 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
+ distribution it is wished to draw approximate samples from.
+ """
+
+[docs] @abstractmethod
+ def h2_flow(self, state: ChainState, dt: ScalarLike):
+ """Apply exact flow map corresponding to `h2` Hamiltonian component.
+
+ `state` argument is modified in place.
+
+ 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.
+ """
+
+
+[docs]class EuclideanMetricSystem(TractableFlowSystem):
r"""Hamiltonian system with a Euclidean metric on the position space.
Here Euclidean metric is defined to mean a metric with a fixed positive definite
@@ -375,30 +423,9 @@ Source code for mici.systems
return self.dh1_dpos(state)
[docs] def h2_flow(self, state: ChainState, dt: ScalarLike):
- """Apply exact flow map corresponding to `h2` Hamiltonian component.
-
- `state` argument is modified in place.
-
- Args:
- state: State to start flow at.
- dt: Time interval to simulate flow for.
- """
state.pos += dt * self.dh2_dmom(state)
[docs] def dh2_flow_dmom(self, dt: ScalarLike) -> tuple[matrices.Matrix, matrices.Matrix]:
- """Derivatives of `h2_flow` flow map with respect to input momentum.
-
- Args:
- dt: Time interval flow simulated for.
-
- Returns:
- dpos_dmom: Matrix representing derivative (Jacobian) of position output of
- `h2_flow` with respect to the value of the momentum component of the
- initial input state.
- dmom_dmom: 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 (dt * self.metric.inv, matrices.IdentityMatrix(self.metric.shape[0]))
[docs] def sample_momentum(self, state: ChainState, rng: Generator) -> ArrayLike:
@@ -532,7 +559,7 @@ 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:`q` 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.
@@ -563,7 +590,7 @@ Source code for mici.systems
Due to the requirement to enforce the constraints on the position and momentum, a
constraint-preserving numerical integrator needs to be used when simulating the
Hamiltonian dynamic associated with the system, e.g.
- `mici.integrators.ConstrainedLeapfrogIntegrator`.
+ :py:class:`mici.integrators.ConstrainedLeapfrogIntegrator`.
References:
@@ -584,80 +611,75 @@ Source code for mici.systems
grad_neg_log_dens: Optional[GradientFunction] = None,
jacob_constr: Optional[JacobianFunction] = None,
):
- """
+ r"""
Args:
neg_log_dens: Function which given a position array returns the negative
logarithm of an unnormalized probability density on the constrained
position space with respect to the Hausdorff measure on the constraint
- manifold (if `dens_wrt_hausdorff == True`) or alternatively the negative
- logarithm of an unnormalized probability density on the unconstrained
- (ambient) position space with respect to the Lebesgue measure. In the
- former case the target distribution it is wished to draw approximate
- samples from is assumed to be directly specified by the density function
- on the manifold. In the latter case the density function is instead
- taken to specify a prior distribution on the ambient space with the
- target distribution then corresponding to the posterior distribution
- when conditioning on the (zero Lebesgue measure) event `constr(pos) ==
- 0`. This target posterior distribution has support on the differentiable
- manifold implicitly defined by the constraint equation, with density
- with respect to the Hausdorff measure on the manifold corresponding to
- the ratio of the prior density (specified by `neg_log_dens`) and the
+ manifold (if :code:`dens_wrt_hausdorff == True`) or alternatively the
+ negative logarithm of an unnormalized probability density on the
+ unconstrained (ambient) position space with respect to the Lebesgue
+ measure. In the former case the target distribution it is wished to draw
+ approximate samples from is assumed to be directly specified by the
+ density function on the manifold. In the latter case the density
+ function is instead taken to specify a prior distribution on the ambient
+ space with the target distribution then corresponding to the posterior
+ distribution when conditioning on the (zero Lebesgue measure) event
+ :code:`constr(q) == 0` where :code:`q` is the position array. This
+ target posterior distribution has support on the differentiable manifold
+ implicitly defined by the constraint equation, with density with respect
+ to the Hausdorff measure on the manifold corresponding to the ratio of
+ the prior density (specified by :code:`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
-
- where `jacob_constr` is the Jacobian of the constraint function `constr`
- and `metric` is the matrix representation of the metric on the ambient
- space.
+ :code:`gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T`
+ where :code:`jacob_constr` is the Jacobian of the constraint function
+ :code:`constr` and :code:`metric` is the matrix representation of the
+ metric on the ambient space.
constr: Function which given a position rray return as a 1D array the value
of the (vector-valued) constraint function, the zero level-set of which
implicitly defines the manifold the dynamic is simulated on.
metric: Matrix object corresponding to matrix representation of metric on
*unconstrained* position space and covariance of Gaussian marginal
- distribution on *unconstrained* momentum vector. If `None` is passed
- (the default), the identity matrix will be used. If a 1D array is passed
- then this is assumed to specify a metric with positive diagonal matrix
- representation and the array the matrix diagonal. If a 2D array is
- passed then this is assumed to specify a metric with a dense positive
+ distribution on *unconstrained* momentum vector. If :code:`None` is
+ passed (the default), the identity matrix will be used. If a 1D array is
+ passed then this is assumed to specify a metric with positive diagonal
+ matrix representation and the array the matrix diagonal. If a 2D array
+ is passed then this is assumed to specify a metric with a dense positive
definite matrix representation specified by the array. Otherwise if the
- value is a `mici.matrices.PositiveDefiniteMatrix` subclass it is assumed
- to directly specify the metric matrix representation.
- dens_wrt_hausdorff: Whether the `neg_log_dens` function specifies the
+ value is a :py:class:`mici.matrices.PositiveDefiniteMatrix` subclass it
+ is assumed to directly specify the metric matrix representation.
+ dens_wrt_hausdorff: Whether the :code:`neg_log_dens` function specifies the
(negative logarithm) of the density of the target distribution with
- respect to the Hausdorff measure on the manifold directly (True) or
- alternatively the negative logarithm of a density of a prior
+ respect to the Hausdorff measure on the manifold directly (:code:`True`)
+ or alternatively the negative logarithm of a density of a prior
distriubtion on the unconstrained (ambient) position space with respect
to the Lebesgue measure, with the target distribution then corresponding
- to the posterior distribution when conditioning on the event `const(pos)
- == 0` (False). Note that in the former case the base Hausdorff measure
- on the manifold depends on the metric defined on the ambient space, with
- the Hausdorff measure being defined with respect to the metric induced
- on the manifold from this ambient metric.
+ to the posterior distribution when conditioning on the event
+ :code:`constr(pos) == 0` (:code:`False`). Note that in the former case
+ the base Hausdorff measure on the manifold depends on the metric defined
+ on the ambient space, with the Hausdorff measure being defined with
+ respect to the metric induced on the manifold from this ambient metric.
grad_neg_log_dens: Function which given a position array returns the
- derivative of `neg_log_dens` with respect to the position array
+ derivative of :code:`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 a function
- to compute the derivative (and value) of `neg_log_dens` automatically.
+ second being the value of the :code:`neg_log_dens` evaluated at the
+ passed position array. If :code:`None` is passed (the default) an
+ automatic differentiation fallback will be used to attempt to construct
+ a function to compute the derivative (and value) of :code:`neg_log_dens`
+ automatically.
jacob_constr: Function which given a position array computes the Jacobian
(matrix / 2D array of partial derivatives) of the output of the
- constraint function `c = constr(q)` with respect to the position array
- argument `q`, returning the computed Jacobian as a 2D array `jacob` with
-
- .. code-block::
-
- jacob[i, j] = ∂c[i] / ∂q[j]
-
- Optionally the function may instead return a 2-tuple of values with the
- first being the array corresponding to the Jacobian and the second being
- the value of `constr` evaluated at the passed position array. If `None`
+ constraint function :code:`c = constr(q)` with respect to the position
+ array argument :code:`q`, returning the computed Jacobian as a 2D array
+ :code:`jacob` with :code:`jacob[i, j] = ∂c[i] / ∂q[j]`. Optionally the
+ function may instead return a 2-tuple of values with the first being the
+ array corresponding to the Jacobian and the second being the value of
+ :code:`constr` evaluated at the passed position array. If :code:`None`
is passed (the default) an automatic differentiation fallback will be
used to attempt to construct a function to compute the Jacobian (and
- value) of `constr` automatically.
+ value) of :code:`constr`
+ automatically.
"""
super().__init__(
neg_log_dens=neg_log_dens,
@@ -678,7 +700,7 @@ Source code for mici.systems
state: State to compute value at.
Returns:
- Value of `constr(state.pos)` as 1D array.
+ Value of :code:`constr(state.pos)` as 1D array.
"""
return self._constr(state.pos)
@@ -690,7 +712,7 @@ Source code for mici.systems
state: State to compute value at.
Returns:
- Value of Jacobian of `constr(state.pos)` as 2D array.
+ Value of Jacobian of :code:`constr(state.pos)` as 2D array.
"""
return self._jacob_constr(state.pos)
@@ -703,16 +725,16 @@ Source code for mici.systems
) -> MatrixLike:
"""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`.
+ 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
- `jacob_constr_1` if set to `None`.
+ :code:`jacob_constr_1` if set to :code:`None`.
Returns
Object corresponding to computed inner products of the constraint Jacobian
@@ -725,12 +747,13 @@ Source code for mici.systems
The Gram matrix as a position `q` is defined as
- .. code-block::
+ .. code::
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
- where `jacob_constr` is the Jacobian of the constraint function `constr` and
- `metric` is the matrix representation of the metric on the ambient space.
+ where :code:`jacob_constr` is the Jacobian of the constraint function
+ :code:`constr` and :code:`metric` is the matrix representation of the metric on
+ the ambient space.
Args:
state: State to compute value at.
@@ -759,13 +782,14 @@ Source code for mici.systems
[docs] @abstractmethod
def grad_log_det_sqrt_gram(self, state: ChainState) -> ArrayLike:
- """Derivative of (half of) log-determinant of Gram matrix wrt position.
+ """Derivative of half of log-determinant of Gram matrix with respect to position
Args:
state: State to compute value at.
Returns:
- Value of `log_det_sqrt_gram(state)` derivative with respect to `state.pos`.
+ Value of :code:`log_det_sqrt_gram(state)` derivative with respect to
+ :code:`state.pos`.
"""
[docs] def h1(self, state: ChainState) -> ScalarLike:
@@ -791,7 +815,7 @@ Source code for mici.systems
co-tangent space of.
Returns:
- Projected momentum in the co-tangent space at `state.pos`.
+ Projected momentum in the co-tangent space at :code:`state.pos`.
"""
# Use parenthesis to force right-to-left evaluation to avoid
# matrix-matrix products
@@ -809,7 +833,8 @@ Source code for mici.systems
[docs]class DenseConstrainedEuclideanMetricSystem(ConstrainedEuclideanMetricSystem):
r"""Euclidean Hamiltonian system subject to a dense set of constraints.
- See `ConstrainedEuclideanMetricSystem` for more details about constrained systems.
+ See :py:class:`ConstrainedEuclideanMetricSystem` for more details about constrained
+ systems.
"""
def __init__(
@@ -827,100 +852,88 @@ Source code for mici.systems
neg_log_dens: Function which given a position array returns the negative
logarithm of an unnormalized probability density on the constrained
position space with respect to the Hausdorff measure on the constraint
- manifold (if `dens_wrt_hausdorff == True`) or alternatively the negative
- logarithm of an unnormalized probability density on the unconstrained
- (ambient) position space with respect to the Lebesgue measure. In the
- former case the target distribution it is wished to draw approximate
- samples from is assumed to be directly specified by the density function
- on the manifold. In the latter case the density function is instead
- taken to specify a prior distribution on the ambient space with the
- target distribution then corresponding to the posterior distribution
- when conditioning on the (zero Lebesgue measure) event `constr(pos) ==
- 0`. This target posterior distribution has support on the differentiable
- manifold implicitly defined by the constraint equation, with density
- with respect to the Hausdorff measure on the manifold corresponding to
- the ratio of the prior density (specified by `neg_log_dens`) and the
+ manifold (if :code:`dens_wrt_hausdorff == True`) or alternatively the
+ negative logarithm of an unnormalized probability density on the
+ unconstrained (ambient) position space with respect to the Lebesgue
+ measure. In the former case the target distribution it is wished to draw
+ approximate samples from is assumed to be directly specified by the
+ density function on the manifold. In the latter case the density
+ function is instead taken to specify a prior distribution on the ambient
+ space with the target distribution then corresponding to the posterior
+ distribution when conditioning on the (zero Lebesgue measure) event
+ :code:`constr(q) == 0` where :code:`q` is the position array. This
+ target posterior distribution has support on the differentiable manifold
+ implicitly defined by the constraint equation, with density with respect
+ to the Hausdorff measure on the manifold corresponding to the ratio of
+ the prior density (specified by :code:`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
-
- where `jacob_constr` is the Jacobian of the constraint function `constr`
- and `metric` is the matrix representation of the metric on the ambient
- space.
- constr: Function which given a position array return as a 1D array the value
+ :code:`gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T`
+ where :code:`jacob_constr` is the Jacobian of the constraint function
+ :code:`constr` and :code:`metric` is the matrix representation of the
+ metric on the ambient space.
+ constr: Function which given a position rray return as a 1D array the value
of the (vector-valued) constraint function, the zero level-set of which
implicitly defines the manifold the dynamic is simulated on.
metric: Matrix object corresponding to matrix representation of metric on
*unconstrained* position space and covariance of Gaussian marginal
- distribution on *unconstrained* momentum vector. If `None` is passed
- (the default), the identity matrix will be used. If a 1D array is passed
- then this is assumed to specify a metric with positive diagonal matrix
- representation and the array the matrix diagonal. If a 2D array is
- passed then this is assumed to specify a metric with a dense positive
+ distribution on *unconstrained* momentum vector. If :code:`None` is
+ passed (the default), the identity matrix will be used. If a 1D array is
+ passed then this is assumed to specify a metric with positive diagonal
+ matrix representation and the array the matrix diagonal. If a 2D array
+ is passed then this is assumed to specify a metric with a dense positive
definite matrix representation specified by the array. Otherwise if the
- value is a `mici.matrices.PositiveDefiniteMatrix` subclass it is assumed
- to directly specify the metric matrix representation.
- dens_wrt_hausdorff: Whether the `neg_log_dens` function specifies the
+ value is a :py:class:`mici.matrices.PositiveDefiniteMatrix` subclass it
+ is assumed to directly specify the metric matrix representation.
+ dens_wrt_hausdorff: Whether the :code:`neg_log_dens` function specifies the
(negative logarithm) of the density of the target distribution with
- respect to the Hausdorff measure on the manifold directly (True) or
- alternatively the negative logarithm of a density of a prior
+ respect to the Hausdorff measure on the manifold directly (:code:`True`)
+ or alternatively the negative logarithm of a density of a prior
distriubtion on the unconstrained (ambient) position space with respect
to the Lebesgue measure, with the target distribution then corresponding
- to the posterior distribution when conditioning on the event `const(pos)
- == 0` (False). Note that in the former case the base Hausdorff measure
- on the manifold depends on the metric defined on the ambient space, with
- the Hausdorff measure being defined with respect to the metric induced
- on the manifold from this ambient metric.
+ to the posterior distribution when conditioning on the event
+ :code:`constr(pos) == 0` (:code:`False`). Note that in the former case
+ the base Hausdorff measure on the manifold depends on the metric defined
+ on the ambient space, with the Hausdorff measure being defined with
+ respect to the metric induced on the manifold from this ambient metric.
grad_neg_log_dens: Function which given a position array returns the
- derivative of `neg_log_dens` with respect to the position array
+ derivative of :code:`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 a function
- to compute the derivative (and value) of `neg_log_dens` automatically.
+ second being the value of the :code:`neg_log_dens` evaluated at the
+ passed position array. If :code:`None` is passed (the default) an
+ automatic differentiation fallback will be used to attempt to construct
+ a function to compute the derivative (and value) of :code:`neg_log_dens`
+ automatically.
jacob_constr: Function which given a position array computes the Jacobian
(matrix / 2D array of partial derivatives) of the output of the
- constraint function `c = constr(q)` with respect to the position array
- argument `q`, returning the computed Jacobian as a 2D array `jacob` with
-
- .. code-block::
-
- jacob[i, j] = ∂c[i] / ∂q[j]
-
- Optionally the function may instead return a 2-tuple of values with the
- first being the array corresponding to the Jacobian and the second being
- the value of `constr` evaluated at the passed position array. If `None`
+ constraint function :code:`c = constr(q)` with respect to the position
+ array argument :code:`q`, returning the computed Jacobian as a 2D array
+ :code:`jacob` with :code:`jacob[i, j] = ∂c[i] / ∂q[j]`. Optionally the
+ function may instead return a 2-tuple of values with the first being the
+ array corresponding to the Jacobian and the second being the value of
+ :code:`constr` evaluated at the passed position array. If :code:`None`
is passed (the default) an automatic differentiation fallback will be
used to attempt to construct a function to compute the Jacobian (and
- value) of `neg_log_dens` automatically.
+ value) of :code:`constr`
+ automatically.
mhp_constr: Function which given a position array returns another function
which takes a 2D array as an argument and returns the
- *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))
-
- where `hess` is the `(dim_constr, dim_pos, dim_pos)` shaped
- vector-Hessian of `c = constr(q)` with respect to `q` i.e. the array of
- second-order partial derivatives of such that
-
- .. code-block::
-
- hess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k])
-
- Optionally the function may instead return a 3-tuple of values with the
- first a function to compute a MHP of `constr`, the second a 2D array
- corresponding to the Jacobian of `constr`, and the third the value of
- `constr`, all evaluated at the passed position array. If `None` is
- passed (the default) an automatic differentiation fallback will be used
- to attempt to construct a function which calculates the MHP (and
- Jacobian and value) of `constr` automatically.
+ *matrix-Hessian-product* (MHP) of the constraint function :code:`constr`
+ with respect to the position array argument. The MHP is here defined as
+ a function of a :code:`(dim_constr, dim_pos)` shaped 2D array :code:`m`
+ as :code:`mhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1))`
+ where :code:`hess` is the :code:`(dim_constr, dim_pos, dim_pos)` shaped
+ vector-Hessian of :code:`c = constr(q)` with respect to :code:`q` i.e.
+ the array of second-order partial derivatives of such that
+ :code:`hess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k])`. Optionally the function
+ may instead return a 3-tuple of values with the first a function to
+ compute a MHP of :code:`constr`, the second a 2D array corresponding to
+ the Jacobian of :code:`constr`, and the third the value of
+ :code:`constr`, all evaluated at the passed position array. If
+ :code:`None` is passed (the default) an automatic differentiation
+ fallback will be used to attempt to construct a function which
+ calculates the MHP (and Jacobian and value) of :code:`constr`
+ automatically.
"""
super().__init__(
neg_log_dens=neg_log_dens,
diff --git a/genindex.html b/genindex.html
index 8200f10..9c5dd33 100644
--- a/genindex.html
+++ b/genindex.html
@@ -373,6 +373,8 @@ D
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
dh2_dmom() (mici.systems.CholeskyFactoredRiemannianMetricSystem method)
@@ -399,6 +401,8 @@ D
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
dh2_dpos() (mici.systems.CholeskyFactoredRiemannianMetricSystem method)
@@ -425,6 +429,8 @@ D
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
dh2_flow_dmom() (mici.systems.ConstrainedEuclideanMetricSystem method)
@@ -437,6 +443,8 @@ D
(mici.systems.GaussianDenseConstrainedEuclideanMetricSystem method)
(mici.systems.GaussianEuclideanMetricSystem method)
+
+ (mici.systems.TractableFlowSystem method)
dh_dmom() (mici.systems.CholeskyFactoredRiemannianMetricSystem method)
@@ -463,6 +471,8 @@ D
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
@@ -491,6 +501,8 @@ D
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
diagonal (mici.matrices.BlockColumnMatrix property)
@@ -821,6 +833,8 @@ G
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
@@ -893,6 +907,8 @@ H
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
h1() (mici.systems.CholeskyFactoredRiemannianMetricSystem method)
@@ -919,6 +935,8 @@ H
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
h1_flow() (mici.systems.CholeskyFactoredRiemannianMetricSystem method)
@@ -945,6 +963,8 @@ H
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
@@ -973,6 +993,8 @@ H
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
h2_flow() (mici.systems.ConstrainedEuclideanMetricSystem method)
@@ -985,6 +1007,8 @@ H
(mici.systems.GaussianDenseConstrainedEuclideanMetricSystem method)
(mici.systems.GaussianEuclideanMetricSystem method)
+
+ (mici.systems.TractableFlowSystem method)
HamiltonianDivergenceError
@@ -1553,6 +1577,8 @@ N
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
@@ -1645,8 +1671,6 @@ R
(mici.progressbars.SequenceProgressBar method)
-
- S
(mici.systems.SoftAbsRiemannianMetricSystem method)
(mici.systems.System method)
+
+ (mici.systems.TractableFlowSystem method)
sample_pymc3_model() (in module mici.interop)
@@ -2001,6 +2043,18 @@ S
System (class in mici.systems)
+ system (mici.samplers.DynamicMultinomialHMC property)
+
+
@@ -2092,9 +2146,25 @@ T
TractableFlowIntegrator (class in mici.integrators)
+
+ TractableFlowSystem (class in mici.systems)
Transition (class in mici.transitions)
+ transitions (mici.samplers.DynamicMultinomialHMC property)
+
+
- transpose (mici.matrices.BlockColumnMatrix property)
diff --git a/mici.adapters.html b/mici.adapters.html
index a8363d0..8a033f9 100644
--- a/mici.adapters.html
+++ b/mici.adapters.html
@@ -175,7 +175,7 @@
An adapter which requires only local information to adapt the transition
parameters should be classified as fast while one which requires more global
information and so more chain iterations should be classified as slow i.e.
-is_fast == False.
+is_fast == False.
diff --git a/mici.html b/mici.html
index a5c2557..f11e3f8 100644
--- a/mici.html
+++ b/mici.html
@@ -254,6 +254,7 @@ SubmodulesScalarRiemannianMetricSystem
SoftAbsRiemannianMetricSystemSystemTractableFlowSystem
mici.transitions module
diff --git a/mici.integrators.html b/mici.integrators.html
index 8af50af..c51eb74 100644
--- a/mici.integrators.html
+++ b/mici.integrators.html
@@ -16,6 +16,7 @@
+
@@ -115,17 +116,20 @@
Bases: SymmetricCompositionIntegrator
Four-stage symmetric composition integrator due to Blanes, Casas & Sanz-Serna.
Described in equation (6.8) in Blanes, Casas, Sanz-Serna (2014).
-References:
-Blanes, S., Casas, F., & Sanz-Serna, J. M. (2014).
+
References
+
+Blanes, S., Casas, F., & Sanz-Serna, J. M. (2014).
Numerical integrators for the Hybrid Monte Carlo method.
-SIAM Journal on Scientific Computing, 36(4), A1556-A1580.
+SIAM Journal on Scientific Computing, 36(4), A1556-A1580.
- Parameters:
-system (System) – Hamiltonian system to integrate the 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 (TractableFlowSystem) – Hamiltonian system to integrate the dynamics of with tractable
+Hamiltonian component flows.
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.
@@ -154,17 +158,20 @@
Bases: SymmetricCompositionIntegrator
Three-stage symmetric composition integrator due to Blanes, Casas & Sanz-Serna.
Described in equation (6.7) in Blanes, Casas, Sanz-Serna (2014).
-References:
-Blanes, S., Casas, F., & Sanz-Serna, J. M. (2014).
+
References
+
+Blanes, S., Casas, F., & Sanz-Serna, J. M. (2014).
Numerical integrators for the Hybrid Monte Carlo method.
-SIAM Journal on Scientific Computing, 36(4), A1556-A1580.
+SIAM Journal on Scientific Computing, 36(4), A1556-A1580.
- Parameters:
-system (System) – Hamiltonian system to integrate the 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 (TractableFlowSystem) – Hamiltonian system to integrate the dynamics of with tractable
+Hamiltonian component flows.
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.
@@ -193,17 +200,20 @@
Bases: SymmetricCompositionIntegrator
Two-stage symmetric composition integrator due to Blanes, Casas & Sanz-Serna.
Described in equation (6.4) in Blanes, Casas, Sanz-Serna (2014).
-References:
-Blanes, S., Casas, F., & Sanz-Serna, J. M. (2014).
+
References
+
+Blanes, S., Casas, F., & Sanz-Serna, J. M. (2014).
Numerical integrators for the Hybrid Monte Carlo method.
-SIAM Journal on Scientific Computing, 36(4), A1556-A1580.
+SIAM Journal on Scientific Computing, 36(4), A1556-A1580.
- Parameters:
-system (System) – Hamiltonian system to integrate the 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 (TractableFlowSystem) – Hamiltonian system to integrate the dynamics of with tractable
+Hamiltonian component flows.
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.
@@ -405,24 +415,25 @@
Bases: Integrator
Implicit leapfrog integrator for Hamiltonians with a non-separable component.
Also known as the generalised leapfrog method.
-The Hamiltonian function h is assumed to take the form
-
-h(q, p) = h₁(q) + h₂(q, p)
-
-where q and p are the position and momentum variables respectively, h₁ is a
-Hamiltonian component function for which the exact flow can be computed and h₂ is
-a Hamiltonian component function of the position and momentum variables, which may
-be non-separable and for which exact simulation of the correspond Hamiltonian flow
-may not be possible.
+The Hamiltonian function \(h\) is assumed to take the form
+
+\[h(q, p) = h_1(q) + h_2(q, p)\]
+where \(q\) and \(p\) are the position and momentum variables respectively,
+\(h_1\) is a Hamiltonian component function for which the exact flow can be
+computed and \(h_2\) is a Hamiltonian component function of the position and
+momentum variables, which may be non-separable and for which exact simulation of the
+correspond Hamiltonian flow may not be possible.
A pair of implicit component updates are used to approximate the flow due to the
-h₂ Hamiltonian component, with a fixed-point iteration used to solve the
+\(h_2\) Hamiltonian component, with a fixed-point iteration used to solve the
non-linear system of equations.
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.
-References:
-Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
-Cambridge University Press.
+References
+
+Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
+Cambridge University Press.
- Parameters:
@@ -541,19 +552,17 @@
class mici.integrators.Integrator(system, step_size=None)[source]
Bases: ABC
Base class for integrators for simulating Hamiltonian dynamics.
-For a Hamiltonian function h with position variables q and momentum variables
-p, the canonical Hamiltonian dynamic is defined by the ordinary differential
-equation system
-
-q̇ = ∇₂h(q, p)
-ṗ = -∇₁h(q, p)
-
+For a Hamiltonian function \(h\) with position variables \(q\) and momentum
+variables \(p\), the canonical Hamiltonian dynamic is defined by the ordinary
+differential equation system
+
+\[\dot{q} = \nabla_2 h(q, p), \qquad \dot{p} = -\nabla_1 h(q, p)\]
with the flow map corresponding to the solution of the corresponding initial value
problem a time-reversible and symplectic (and by consequence volume-preserving) map.
-Derived classes implement a step method which approximates the flow-map over some
-small time interval, while conserving the properties of being time-reversible and
-symplectic, with composition of this integrator step method allowing simulation of
-time-discretised trajectories of the Hamiltonian dynamics.
+Derived classes implement a step() method which approximates the flow-map
+over some small time interval, while conserving the properties of being
+time-reversible and symplectic, with composition of this integrator step method
+allowing simulation of time-discretised trajectories of the Hamiltonian dynamics.
- Parameters:
@@ -588,29 +597,29 @@
class mici.integrators.LeapfrogIntegrator(system, step_size=None)[source]
Bases: TractableFlowIntegrator
Leapfrog integrator for Hamiltonian systems with tractable component flows.
-For separable Hamiltonians of the form
-
-h(q, p) = h₁(q) + h₂(p)
-
-where h₁ is the potential energy and h₂ is the kinetic energy, this integrator
-corresponds to the classic (position) Störmer-Verlet method.
+For separable Hamiltonians of the
+
+\[h(q, p) = h_1(q) + h_2(p)\]
+where \(h_1\) is the potential energy and \(h_2\) is the kinetic energy,
+this integrator corresponds to the classic (position) Störmer-Verlet method.
The integrator can also be applied to the more general Hamiltonian splitting
-
-h(q, p) = h₁(q) + h₂(q, p)
-
-providing the flows for h₁ and h₂ are both tractable.
+
+\[h(q, p) = h_1(q) + h_2(q, p)\]
+providing the flows for \(h_1\) and \(h_2\) are both tractable.
For more details see Sections 2.6 and 4.2.2 in Leimkuhler and Reich (2004).
-References:
-Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
-Cambridge University Press.
+References
+
+Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
+Cambridge University Press.
- Parameters:
-system (System) – Hamiltonian system to integrate the dynamics of. Must define both
-h1_flow and h2_flow methods.
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 (TractableFlowSystem) – Hamiltonian system to integrate the dynamics of with tractable
+Hamiltonian component flows.
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.
@@ -637,57 +646,65 @@
-
class mici.integrators.SymmetricCompositionIntegrator(system, free_coefficients, step_size=None, initial_h1_flow_step=True)[source]
Bases: TractableFlowIntegrator
-Symmetric composition integrator for Hamiltonians with tractable component flows.
+Symmetric composition integrator for Hamiltonians with tractable component flows
The Hamiltonian function is assumed to be expressible as the sum of two analytically
tractable components for which the corresponding Hamiltonian flows can be exactly
-simulated. Specifically it is assumed that the Hamiltonian function h takes the form
-
-h(q, p) = h₁(q) + h₂(q, p)
-
-where q and p are the position and momentum variables respectively, and h₁ and
-h₂ are Hamiltonian component functions for which the exact flows, respectively
-Φ₁ and Φ₂, can be computed. An alternating composition can then be formed as
-
-Ψ(t) = A(aₛt) ∘ B(bₛt) ∘ ⋯ ∘ A(a₁t) ∘ B(b₁t) ∘ A(a₀t)
-
-where A = Φ₁ and B = Φ₂ or A = Φ₂ and B = Φ₁, and (a₀, ⋯, aₛ) and
-(b₁, ⋯, bₛ) are a set of coefficients to be determined and s is the number of
-stages in the composition.
+simulated. Specifically it is assumed that the Hamiltonian function \(h\) takes
+the 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 \(h_2\) are Hamiltonian component functions for which the
+exact flows, respectively \(\Phi_1\) and \(\Phi_2\), can be computed. An
+alternating composition can then be formed as
+
+\[\Psi(t) =
+A(a_S t) \circ B(b_S t) \circ \dots \circ A(a_1 t) \circ B(b_1 t) \circ A(a_0 t)\]
+where \(A = \Phi_1\) and \(B = \Phi_2\) or \(A = \Phi_2\) and \(B =
+\Phi_1\), and \((a_0,\dots, a_S)\) and \((b_1, \dots, b_S)\) are a set of
+coefficients to be determined and \(S\) is the number of stages in the
+composition.
To ensure a consistency (i.e. the integrator is at least order one) we require that
-
-sum(a₀, ⋯, aₛ) = sum(b₁, ⋯, bₛ) = 1
-
+
+\[\sum_{s=0}^S a_s = \sum_{s=1}^S b_s = 1.\]
For symmetric compositions we restrict that
-
-aₛ₋ₘ = aₘ and bₛ₊₁₋ₘ = bₛ
-
+
+\[a_{S-m} = a_m, \quad b_{S+1-m} = b_s\]
with symmetric consistent methods of at least order two.
-The combination of the symmetry and consistency requirements mean that a s-stage
-symmetric composition method can be described by (s - 1) ‘free’ coefficients
-
-(a₀, b₁, a₁, ⋯, aₖ, bₖ) with k = (s - 1) / 2 if s is odd or
-(a₀, b₁, a₁, ⋯, aₖ) with k = (s - 2) / 2 if s is even
-
+The combination of the symmetry and consistency requirements mean that a
+\(S\)-stage symmetric composition method can be described by \(S - 1\)
+‘free’ coefficients
+
+\[(a_0, b_1, a_1, \dots, a_K, b_K)\]
+with \(K = (S - 1) / 2\) if \(S\) is odd or
+
+\[(a_0, b_1, a_1, \dots, a_K)\]
+with \(K = (S - 2) / 2\) if \(S\) is even.
The Störmer-Verlet ‘leapfrog’ integrator is a special case corresponding to the
unique (symmetric and consistent) 1-stage integrator.
For more details see Section 6.2 in Leimkuhler and Reich (2004).
-References:
-Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
-Cambridge University Press.
+References
+
+Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14).
+Cambridge University Press.
- Parameters:
-system (System) – Hamiltonian system to integrate the dynamics of.
free_coefficients (Sequence[float]) – Sequence of s - 1 scalar values, where s is the
-number of stages in the symmetric composition, specifying the free
-coefficients (a₀, b₁, a₁, ⋯, aₖ, bₖ) with k = (s - 1) / 2 if s is
-odd or (a₀, b₁, a₁, ⋯, aₖ) with k = (s - 2) / 2 if s is even.
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.
initial_h1_flow_step (bool) – Whether the initial ‘A’ flow in the composition should
-correspond to the flow of the h₁ Hamiltonian component (True) or to
-the flow of the h₂ component (False).
system (TractableFlowSystem) – Hamiltonian system to integrate the dynamics of with tractable
+Hamiltonian component flows.
free_coefficients (Sequence[float]) – Sequence of \(S - 1\) scalar values, where \(S\)
+is the number of stages in the symmetric composition, specifying the
+free coefficients \((a_0, b_1, a_1, \dots, a_K, b_K)\) with \(K
+= (S - 1) / 2\) if \(S\) is odd or \((a_0, b_1, a_1, \dots,
+a_K)\) with \(k = (S - 2) / 2\) if S is even.
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.
initial_h1_flow_step (bool) – Whether the initial \(A\) flow in the composition
+should correspond to the flow of the h_1 Hamiltonian component
+(True) or to the flow of the \(h_2\) component
+(False).
@@ -717,21 +734,21 @@
Base class for integrators for Hamiltonian systems with tractable component flows
The Hamiltonian function is assumed to be expressible as the sum of two analytically
tractable components for which the corresponding Hamiltonian flows can be exactly
-simulated. Specifically it is assumed that the Hamiltonian function h takes the
-form
-
-h(q, p) = h₁(q) + h₂(q, p)
-
-where q and p are the position and momentum variables respectively, and h₁ and
-h₂ are Hamiltonian component functions for which the exact flows can be computed.
+simulated. Specifically it is assumed that the Hamiltonian function \(h\) takes
+the 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 \(h_2\) are Hamiltonian component functions for which the
+exact flows can be computed.
- Parameters:
-system (System) – Hamiltonian system to integrate the dynamics of. Must define both
-h1_flow and h2_flow methods.
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 (TractableFlowSystem) – Hamiltonian system to integrate the dynamics of with tractable
+Hamiltonian component flows.
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.
diff --git a/mici.samplers.html b/mici.samplers.html
index 29a203f..cea524e 100644
--- a/mici.samplers.html
+++ b/mici.samplers.html
@@ -186,6 +186,12 @@
Maximum depth to expand trajectory binary tree to.
+
+-
+property rng: Generator
+NumPy random number generator object.
+
+
-
sample_chains(n_warm_up_iter, n_main_iter, init_states, **kwargs)
@@ -298,6 +304,18 @@
+
+-
+property system: System
+Hamiltonian system corresponding to joint distribution on augmented space.
+
+
+
+-
+property transitions: dict[str, Transition]
+Dictionary of transition kernels sampled from in each chain iteration.
+
+
@@ -376,6 +394,12 @@
Maximum depth to expand trajectory binary tree to.
+
+-
+property rng: Generator
+NumPy random number generator object.
+
+
-
sample_chains(n_warm_up_iter, n_main_iter, init_states, **kwargs)
@@ -488,6 +512,18 @@
+-
+property system: System
+Hamiltonian system corresponding to joint distribution on augmented space.
+
+
+
+-
+property transitions: dict[str, Transition]
+Dictionary of transition kernels sampled from in each chain iteration.
+
+
@@ -551,22 +587,30 @@
- Parameters:
-simulated. (system Hamiltonian system to be) –
simulated (system Hamiltonian system to be) – distribution on augmented space.
joint (corresponding to) – distribution on augmented space.
rng (Generator) – Numpy random number generator.
integration_transition (IntegrationTransition) – Markov transition kernel which leaves canonical
+
integration_transition (IntegrationTransition) – Markov transition kernel which leaves joint
distribution invariant and jointly updates the position and momentum
components of the chain state by integrating the Hamiltonian dynamics of
the system to propose new values for the state.
momentum_transition (Optional[MomentumTransition]) – Markov transition kernel which leaves the conditional
-distribution on the momentum under the canonical distribution invariant,
+distribution on the momentum under the join distribution invariant,
updating only the momentum component of the chain state. If set to
-None the momentum transition operator
-mici.transitions.IndependentMomentumTransition will be used, which
-independently samples the momentum from its conditional distribution.
None the momentum transition operator
+mici.transitions.IndependentMomentumTransition will be used,
+which independently samples the momentum from its conditional
+distribution.
system (System) –
+-
+property rng: Generator
+NumPy random number generator object.
+
+
+
+-
+property system: System
+Hamiltonian system corresponding to joint distribution on augmented space.
+
+
+
+-
+property transitions: dict[str, Transition]
+Dictionary of transition kernels sampled from in each chain iteration.
+
+
@@ -730,6 +786,12 @@
+
+-
+property rng: Generator
+NumPy random number generator object.
+
+
-
sample_chains(n_warm_up_iter, n_main_iter, init_states, *, trace_funcs=None, adapters=None, stager=None, n_process=1, trace_warm_up=False, max_threads_per_process=None, force_memmap=False, memmap_path=None, monitor_stats=None, display_progress=True, progress_bar_class=None)[source]
@@ -836,6 +898,12 @@
+
+-
+property transitions: dict[str, Transition]
+Dictionary of transition kernels sampled from in each chain iteration.
+
+
@@ -890,6 +958,12 @@
Interval to uniformly draw number of integrator steps from.
+
+-
+property rng: Generator
+NumPy random number generator object.
+
+
-
sample_chains(n_warm_up_iter, n_main_iter, init_states, **kwargs)
@@ -1002,6 +1076,18 @@
+
+-
+property system: System
+Hamiltonian system corresponding to joint distribution on augmented space.
+
+
+
+-
+property transitions: dict[str, Transition]
+Dictionary of transition kernels sampled from in each chain iteration.
+
+
@@ -1051,6 +1137,12 @@
Number of integrator steps per integrator transition.
+
+-
+property rng: Generator
+NumPy random number generator object.
+
+
-
sample_chains(n_warm_up_iter, n_main_iter, init_states, **kwargs)
@@ -1163,6 +1255,18 @@
+
+-
+property system: System
+Hamiltonian system corresponding to joint distribution on augmented space.
+
+
+
+-
+property transitions: dict[str, Transition]
+Dictionary of transition kernels sampled from in each chain iteration.
+
+
diff --git a/mici.stagers.html b/mici.stagers.html
index 4daf9d9..2b9049c 100644
--- a/mici.stagers.html
+++ b/mici.stagers.html
@@ -102,7 +102,7 @@
mici.stagers module
-Classes for controlling sampling of Markov chains split into stages.
+Classes for staging sampling of Markov chains.
-
class mici.stagers.ChainStage(n_iter, adapters, trace_funcs, record_stats)[source]
@@ -113,8 +113,8 @@
n_iter (int) – Number of iterations in chain stage.
adapters (dict[str, Iterable[Adapter]]) – Dictionary of adapters to apply to each transition in chain stage.
trace_funcs (tuple[TraceFunction]) – Functions defining chain variables to trace during chain stage.
record_stats (bool) – Whether to record statistics and traces during chain stage.
trace_funcs (Optional[tuple[TraceFunction]]) – Functions defining chain variables to trace during chain stage.
record_stats (bool) – Whether to record statistics during chain stage.
@@ -129,10 +129,10 @@
abstract stages(n_warm_up_iter, n_main_iter, adapters, trace_funcs, trace_warm_up=False)[source]
Create dictionary specifying labels and parameters of chain sampling stages.
-Constructs a (ordered) dictionary with entries corresponding to the sequence of
+
Constructs an ordered dictionary with entries corresponding to the sequence of
sampling stages when running chains with one or more initial adaptation stages.
The keys of each entry are a string label for the sampling stage and the value a
-ChainStage named tuple specifying the parameters of the stage.
+ChainStage named tuple specifying the parameters of the stage.
- Parameters:
@@ -151,16 +151,19 @@
stored. By default chains are only traced in the iterations of the final
non-adaptive stage - this behaviour can be altered using the
trace_warm_up argument.
-adapters (dict[str, Iterable[Adapter]]) – Dictionary of iterables of Adapter instances keyed by strings
-corresponding to the key of the transition in the sampler transitions
+
adapters (dict[str, Iterable[Adapter]]) – Dictionary of iterables of mici.adapters.Adapter
+instances keyed by strings corresponding to the key of the transition in
+the sampler
+mici.samplers.MarkovChainMonteCarloMethod.transitions
dictionary to apply the adapters to, to use to adaptatively set
parameters of the transitions during the adaptive stages of the chains.
Note that the adapter updates are applied in the order the adapters
appear in the iterables and so if multiple adapters change the same
parameter(s) the order will matter.
trace_warm_up (bool) – Whether to record chain traces and statistics during warm-up
-stage iterations (True) or only record traces and statistics in the
-iterations of the final non-adaptive stage (False, the default).
True) or only record traces and statistics in
+the iterations of the final non-adaptive stage (False, the
+default).
- Returns:
@@ -192,10 +195,10 @@
-
stages(n_warm_up_iter, n_main_iter, adapters, trace_funcs, trace_warm_up=False)[source]
Create dictionary specifying labels and parameters of chain sampling stages.
-Constructs a (ordered) dictionary with entries corresponding to the sequence of
+
Constructs an ordered dictionary with entries corresponding to the sequence of
sampling stages when running chains with one or more initial adaptation stages.
The keys of each entry are a string label for the sampling stage and the value a
-ChainStage named tuple specifying the parameters of the stage.
+ChainStage named tuple specifying the parameters of the stage.
- Parameters:
@@ -204,7 +207,7 @@
iterations may be split between one or more adaptive stages.
n_main_iter (int) – Number of iterations (samples to draw) per chain during main
(non-adaptive) sampling stage.
trace_funcs (Iterable[Callable[[ChainState], dict[str, ArrayLike]]]) – Iterable of functions which compute the variables to be
+
trace_funcs (Iterable[TraceFunction]) – Iterable of functions which compute the variables to be
recorded at each chain iteration, with each trace function being passed
the current state and returning a dictionary of scalar or array values
corresponding to the variable(s) to be stored. The keys in the returned
@@ -214,16 +217,19 @@
stored. By default chains are only traced in the iterations of the final
non-adaptive stage - this behaviour can be altered using the
trace_warm_up argument.
adapters (dict[str, Iterable[Adapter]]) – Dictionary of iterables of Adapter instances keyed by strings
-corresponding to the key of the transition in the sampler transitions
+
adapters (dict[str, Iterable[Adapter]]) – Dictionary of iterables of mici.adapters.Adapter
+instances keyed by strings corresponding to the key of the transition in
+the sampler
+mici.samplers.MarkovChainMonteCarloMethod.transitions
dictionary to apply the adapters to, to use to adaptatively set
parameters of the transitions during the adaptive stages of the chains.
Note that the adapter updates are applied in the order the adapters
appear in the iterables and so if multiple adapters change the same
parameter(s) the order will matter.
trace_warm_up (bool) – Whether to record chain traces and statistics during warm-up
-stage iterations (True) or only record traces and statistics in the
-iterations of the final non-adaptive stage (False, the default).
True) or only record traces and statistics in
+the iterations of the final non-adaptive stage (False, the
+default).
- Returns:
@@ -249,7 +255,8 @@
information. The adapters to be used in both the fast and slow adaptation stages
will be referred to as the fast adapters and the adapters to use in only the slow
adaptation stages the slow adapters. Each adapter self identifies if it is a fast
-adapter by whether the is_fast attribute is set to True.
+adapter by whether the mici.adapters.Adapter.is_fast attribute is set to
+True.
The adaptive warm up iterations are split into three stages:
@@ -298,10 +305,10 @@
-
stages(n_warm_up_iter, n_main_iter, adapters, trace_funcs, trace_warm_up=False)[source]
Create dictionary specifying labels and parameters of chain sampling stages.
-Constructs a (ordered) dictionary with entries corresponding to the sequence of
+
Constructs an ordered dictionary with entries corresponding to the sequence of
sampling stages when running chains with one or more initial adaptation stages.
The keys of each entry are a string label for the sampling stage and the value a
-ChainStage named tuple specifying the parameters of the stage.
+ChainStage named tuple specifying the parameters of the stage.
- Parameters:
@@ -320,16 +327,19 @@
stored. By default chains are only traced in the iterations of the final
non-adaptive stage - this behaviour can be altered using the
trace_warm_up argument.
-adapters (dict[str, Iterable[Adapter]]) – Dictionary of iterables of Adapter instances keyed by strings
-corresponding to the key of the transition in the sampler transitions
+
adapters (dict[str, Iterable[Adapter]]) – Dictionary of iterables of mici.adapters.Adapter
+instances keyed by strings corresponding to the key of the transition in
+the sampler
+mici.samplers.MarkovChainMonteCarloMethod.transitions
dictionary to apply the adapters to, to use to adaptatively set
parameters of the transitions during the adaptive stages of the chains.
Note that the adapter updates are applied in the order the adapters
appear in the iterables and so if multiple adapters change the same
parameter(s) the order will matter.
trace_warm_up (bool) – Whether to record chain traces and statistics during warm-up
-stage iterations (True) or only record traces and statistics in the
-iterations of the final non-adaptive stage (False, the default).
True) or only record traces and statistics in
+the iterations of the final non-adaptive stage (False, the
+default).
- Returns:
diff --git a/mici.systems.html b/mici.systems.html
index f539dab..28061fc 100644
--- a/mici.systems.html
+++ b/mici.systems.html
@@ -73,6 +73,7 @@
ScalarRiemannianMetricSystemSoftAbsRiemannianMetricSystemSystemTractableFlowSystem- mici.transitions module
@@ -456,7 +457,7 @@
\[h_2(q, p) = \frac{1}{2} p^T M^{-1} p.\]
The time-derivative of the constraint equation implies a further set of constraints
-on the momentum \(q\) with :math:` partial c(q) M^{-1} p = 0` at all time
+on the momentum \(q\) 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
@@ -479,7 +480,7 @@
Due to the requirement to enforce the constraints on the position and momentum, a
constraint-preserving numerical integrator needs to be used when simulating the
Hamiltonian dynamic associated with the system, e.g.
-mici.integrators.ConstrainedLeapfrogIntegrator.
+mici.integrators.ConstrainedLeapfrogIntegrator.
References
Lelièvre, T., Rousset, M. and Stoltz, G., 2019. Hybrid Monte Carlo methods for
@@ -492,79 +493,73 @@
- Parameters:
-neg_log_dens (ScalarFunction) –
Function which given a position array returns the negative
+
neg_log_dens (ScalarFunction) – Function which given a position array returns the negative
logarithm of an unnormalized probability density on the constrained
position space with respect to the Hausdorff measure on the constraint
-manifold (if dens_wrt_hausdorff == True) or alternatively the negative
-logarithm of an unnormalized probability density on the unconstrained
-(ambient) position space with respect to the Lebesgue measure. In the
-former case the target distribution it is wished to draw approximate
-samples from is assumed to be directly specified by the density function
-on the manifold. In the latter case the density function is instead
-taken to specify a prior distribution on the ambient space with the
-target distribution then corresponding to the posterior distribution
-when conditioning on the (zero Lebesgue measure) event constr(pos) ==
-0. This target posterior distribution has support on the differentiable
-manifold implicitly defined by the constraint equation, with density
-with respect to the Hausdorff measure on the 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
-gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
-
-
-where jacob_constr is the Jacobian of the constraint function constr
-and metric is the matrix representation of the metric on the ambient
-space.
-dens_wrt_hausdorff == True) or alternatively the
+negative logarithm of an unnormalized probability density on the
+unconstrained (ambient) position space with respect to the Lebesgue
+measure. In the former case the target distribution it is wished to draw
+approximate samples from is assumed to be directly specified by the
+density function on the manifold. In the latter case the density
+function is instead taken to specify a prior distribution on the ambient
+space with the target distribution then corresponding to the posterior
+distribution when conditioning on the (zero Lebesgue measure) event
+constr(q) == 0 where q is the position array. This
+target posterior distribution has support on the differentiable manifold
+implicitly defined by the constraint equation, with density with respect
+to the Hausdorff measure on the 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
+gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
+where jacob_constr is the Jacobian of the constraint function
+constr and metric is the matrix representation of the
+metric on the ambient space.
constr (ArrayFunction) – Function which given a position rray return as a 1D array the value
of the (vector-valued) constraint function, the zero level-set of which
implicitly defines the manifold the dynamic is simulated on.
metric (Optional[MetricLike]) – Matrix object corresponding to matrix representation of metric on
unconstrained position space and covariance of Gaussian marginal
-distribution on unconstrained momentum vector. If None is passed
-(the default), the identity matrix will be used. If a 1D array is passed
-then this is assumed to specify a metric with positive diagonal matrix
-representation and the array the matrix diagonal. If a 2D array is
-passed then this is assumed to specify a metric with a dense positive
+distribution on unconstrained momentum vector. If None is
+passed (the default), the identity matrix will be used. If a 1D array is
+passed then this is assumed to specify a metric with positive diagonal
+matrix representation and the array the matrix diagonal. If a 2D array
+is passed then this is assumed to specify a metric with a dense positive
definite matrix representation specified by the array. Otherwise if the
-value is a mici.matrices.PositiveDefiniteMatrix subclass it is assumed
-to directly specify the metric matrix representation.
dens_wrt_hausdorff (bool) – Whether the neg_log_dens function specifies the
+value is a mici.matrices.PositiveDefiniteMatrix subclass it
+is assumed to directly specify the metric matrix representation.
dens_wrt_hausdorff (bool) – Whether the neg_log_dens function specifies the
(negative logarithm) of the density of the target distribution with
-respect to the Hausdorff measure on the manifold directly (True) or
-alternatively the negative logarithm of a density of a prior
+respect to the Hausdorff measure on the manifold directly (True)
+or alternatively the negative logarithm of a density of a prior
distriubtion on the unconstrained (ambient) position space with respect
to the Lebesgue measure, with the target distribution then corresponding
-to the posterior distribution when conditioning on the event const(pos)
-== 0 (False). Note that in the former case the base Hausdorff measure
-on the manifold depends on the metric defined on the ambient space, with
-the Hausdorff measure being defined with respect to the metric induced
-on the manifold from this ambient metric.
constr(pos) == 0 (False). Note that in the former case
+the base Hausdorff measure on the manifold depends on the metric defined
+on the ambient space, with the Hausdorff measure being defined with
+respect to the metric induced on the manifold from this ambient metric.
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
+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 a function
-to compute the derivative (and value) of neg_log_dens automatically.
jacob_constr (Optional[JacobianFunction]) –
-- Function which given a position array computes the Jacobian
(matrix / 2D array of partial derivatives) of the output of the
-constraint function c = constr(q) with respect to the position array
-argument q, returning the computed Jacobian as a 2D array jacob with
-
- jacob[i, j] = ∂c[i] / ∂q[j]
-
-Optionally the function may instead return a 2-tuple of values with the
-first being the array corresponding to the Jacobian and the second being
-the value of `constr` evaluated at the passed position array. If `None`
+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
+a function to compute the derivative (and value) of neg_log_dens
+automatically.
+
jacob_constr (Optional[JacobianFunction]) – Function which given a position array computes the Jacobian
+(matrix / 2D array of partial derivatives) of the output of the
+constraint function c = constr(q) with respect to the position
+array argument q, returning the computed Jacobian as a 2D array
+jacob with jacob[i, j] = ∂c[i] / ∂q[j]. Optionally the
+function may instead return a 2-tuple of values with the first being the
+array corresponding to the Jacobian and the second being the value of
+constr evaluated at the passed position array. If None
is passed (the default) an automatic differentiation fallback will be
used to attempt to construct a function to compute the Jacobian (and
-value) of `constr` automatically.
-
constr
+automatically.
state (ChainState) – State to compute value at.
- Returns:
-Value of constr(state.pos) as 1D array.
+Value of constr(state.pos) as 1D array.
- Return type:
ArrayLike
@@ -639,24 +634,21 @@
-
dh2_flow_dmom(dt)
-Derivatives of h2_flow flow map with respect to input momentum.
+Derivatives of h2_flow flow map with respect to momentum argument.
- Parameters:
dt (ScalarLike) – Time interval flow simulated for.
- Returns:
-
-- Matrix representing derivative (Jacobian) of position output of
h2_flow with respect to the value of the momentum component of the
-initial input state.
-- dmom_dmom: 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:
-dpos_dmom
+-
@@ -698,13 +690,14 @@
-
abstract grad_log_det_sqrt_gram(state)[source]
-Derivative of (half of) log-determinant of Gram matrix wrt position.
+Derivative of half of log-determinant of Gram matrix with respect to position
- Parameters:
state (ChainState) – State to compute value at.
- Returns:
-Value of log_det_sqrt_gram(state) derivative with respect to state.pos.
+Value of log_det_sqrt_gram(state) derivative with respect to
+state.pos.
- Return type:
ArrayLike
@@ -737,8 +730,9 @@
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
-where jacob_constr is the Jacobian of the constraint function constr and
-metric is the matrix representation of the metric on the ambient space.
+where jacob_constr is the Jacobian of the constraint function
+constr and metric is the matrix representation of the metric on
+the ambient space.
- Parameters:
state (ChainState) – State to compute value at.
@@ -859,7 +853,7 @@
state (ChainState) – State to compute value at.
- Returns:
-Value of Jacobian of constr(state.pos) as 2D array.
+Value of Jacobian of constr(state.pos) as 2D array.
- Return type:
ArrayLike
@@ -871,9 +865,9 @@
-
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.
+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:
@@ -881,7 +875,7 @@
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.
jacob_constr_1 if set to None.
- Return type:
@@ -939,7 +933,7 @@
- Returns:
-Projected momentum in the co-tangent space at state.pos.
+Projected momentum in the co-tangent space at state.pos.
- Return type:
ArrayLike
@@ -974,112 +968,96 @@
class mici.systems.DenseConstrainedEuclideanMetricSystem(neg_log_dens, constr, metric=None, dens_wrt_hausdorff=True, grad_neg_log_dens=None, jacob_constr=None, mhp_constr=None)[source]
Bases: ConstrainedEuclideanMetricSystem
Euclidean Hamiltonian system subject to a dense set of constraints.
-See ConstrainedEuclideanMetricSystem for more details about constrained systems.
+See ConstrainedEuclideanMetricSystem for more details about constrained
+systems.
- Parameters:
-neg_log_dens (ScalarFunction) –
-- Function which given a position array returns the negative
logarithm of an unnormalized probability density on the constrained
+
neg_log_dens (ScalarFunction) – Function which given a position array returns the negative
+logarithm of an unnormalized probability density on the constrained
position space with respect to the Hausdorff measure on the constraint
-manifold (if dens_wrt_hausdorff == True) or alternatively the negative
-logarithm of an unnormalized probability density on the unconstrained
-(ambient) position space with respect to the Lebesgue measure. In the
-former case the target distribution it is wished to draw approximate
-samples from is assumed to be directly specified by the density function
-on the manifold. In the latter case the density function is instead
-taken to specify a prior distribution on the ambient space with the
-target distribution then corresponding to the posterior distribution
-when conditioning on the (zero Lebesgue measure) event constr(pos) ==
-0. This target posterior distribution has support on the differentiable
-manifold implicitly defined by the constraint equation, with density
-with respect to the Hausdorff measure on the 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
-
- gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
-
-where `jacob_constr` is the Jacobian of the constraint function `constr`
-and `metric` is the matrix representation of the metric on the ambient
-space.
-
-
-constr (ArrayFunction) – Function which given a position array return as a 1D array the value
+manifold (if dens_wrt_hausdorff == True) or alternatively the
+negative logarithm of an unnormalized probability density on the
+unconstrained (ambient) position space with respect to the Lebesgue
+measure. In the former case the target distribution it is wished to draw
+approximate samples from is assumed to be directly specified by the
+density function on the manifold. In the latter case the density
+function is instead taken to specify a prior distribution on the ambient
+space with the target distribution then corresponding to the posterior
+distribution when conditioning on the (zero Lebesgue measure) event
+constr(q) == 0 where q is the position array. This
+target posterior distribution has support on the differentiable manifold
+implicitly defined by the constraint equation, with density with respect
+to the Hausdorff measure on the 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
+gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
+where jacob_constr is the Jacobian of the constraint function
+constr and metric is the matrix representation of the
+metric on the ambient space.
constr (ArrayFunction) – Function which given a position rray return as a 1D array the value
of the (vector-valued) constraint function, the zero level-set of which
implicitly defines the manifold the dynamic is simulated on.
metric (Optional[MetricLike]) – Matrix object corresponding to matrix representation of metric on
unconstrained position space and covariance of Gaussian marginal
-distribution on unconstrained momentum vector. If None is passed
-(the default), the identity matrix will be used. If a 1D array is passed
-then this is assumed to specify a metric with positive diagonal matrix
-representation and the array the matrix diagonal. If a 2D array is
-passed then this is assumed to specify a metric with a dense positive
+distribution on unconstrained momentum vector. If None is
+passed (the default), the identity matrix will be used. If a 1D array is
+passed then this is assumed to specify a metric with positive diagonal
+matrix representation and the array the matrix diagonal. If a 2D array
+is passed then this is assumed to specify a metric with a dense positive
definite matrix representation specified by the array. Otherwise if the
-value is a mici.matrices.PositiveDefiniteMatrix subclass it is assumed
-to directly specify the metric matrix representation.
dens_wrt_hausdorff (bool) – Whether the neg_log_dens function specifies the
+value is a mici.matrices.PositiveDefiniteMatrix subclass it
+is assumed to directly specify the metric matrix representation.
dens_wrt_hausdorff (bool) – Whether the neg_log_dens function specifies the
(negative logarithm) of the density of the target distribution with
-respect to the Hausdorff measure on the manifold directly (True) or
-alternatively the negative logarithm of a density of a prior
+respect to the Hausdorff measure on the manifold directly (True)
+or alternatively the negative logarithm of a density of a prior
distriubtion on the unconstrained (ambient) position space with respect
to the Lebesgue measure, with the target distribution then corresponding
-to the posterior distribution when conditioning on the event const(pos)
-== 0 (False). Note that in the former case the base Hausdorff measure
-on the manifold depends on the metric defined on the ambient space, with
-the Hausdorff measure being defined with respect to the metric induced
-on the manifold from this ambient metric.
constr(pos) == 0 (False). Note that in the former case
+the base Hausdorff measure on the manifold depends on the metric defined
+on the ambient space, with the Hausdorff measure being defined with
+respect to the metric induced on the manifold from this ambient metric.
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
+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 a function
-to compute the derivative (and value) of neg_log_dens automatically.
jacob_constr (Optional[JacobianFunction]) –
-- Function which given a position array computes the Jacobian
(matrix / 2D array of partial derivatives) of the output of the
-constraint function c = constr(q) with respect to the position array
-argument q, returning the computed Jacobian as a 2D array jacob with
-
- jacob[i, j] = ∂c[i] / ∂q[j]
-
-Optionally the function may instead return a 2-tuple of values with the
-first being the array corresponding to the Jacobian and the second being
-the value of `constr` evaluated at the passed position array. If `None`
+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
+a function to compute the derivative (and value) of neg_log_dens
+automatically.
+
jacob_constr (Optional[JacobianFunction]) – Function which given a position array computes the Jacobian
+(matrix / 2D array of partial derivatives) of the output of the
+constraint function c = constr(q) with respect to the position
+array argument q, returning the computed Jacobian as a 2D array
+jacob with jacob[i, j] = ∂c[i] / ∂q[j]. Optionally the
+function may instead return a 2-tuple of values with the first being the
+array corresponding to the Jacobian and the second being the value of
+constr evaluated at the passed position array. If None
is passed (the default) an automatic differentiation fallback will be
used to attempt to construct a function to compute the Jacobian (and
-value) of `neg_log_dens` automatically.
-
mhp_constr (Optional[MatrixHessianProductFunction]) –
-- Function which given a position array returns another function
which takes a 2D array as an argument and returns the
-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
-
- mhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1))
-
-where `hess` is the `(dim_constr, dim_pos, dim_pos)` shaped
-vector-Hessian of `c = constr(q)` with respect to `q` i.e. the array of
-second-order partial derivatives of such that
-
-
- hess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k])
-
-Optionally the function may instead return a 3-tuple of values with the
-first a function to compute a MHP of `constr`, the second a 2D array
-corresponding to the Jacobian of `constr`, and the third the value of
-`constr`, all evaluated at the passed position array. If `None` is
-passed (the default) an automatic differentiation fallback will be used
-to attempt to construct a function which calculates the MHP (and
-Jacobian and value) of `constr` automatically.
-
-
-constr
+automatically.
+mhp_constr (Optional[MatrixHessianProductFunction]) – Function which given a position array returns another function
+which takes a 2D array as an argument and returns the
+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
+as mhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1))
+where hess is the (dim_constr, dim_pos, dim_pos) shaped
+vector-Hessian of c = constr(q) with respect to q i.e.
+the array of second-order partial derivatives of such that
+hess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k]). Optionally the function
+may instead return a 3-tuple of values with the first a function to
+compute a MHP of constr, the second a 2D array corresponding to
+the Jacobian of constr, and the third the value of
+constr, all evaluated at the passed position array. If
+None is passed (the default) an automatic differentiation
+fallback will be used to attempt to construct a function which
+calculates the MHP (and Jacobian and value) of constr
+automatically.
@@ -1092,7 +1070,7 @@
state (ChainState) – State to compute value at.
- Returns:
-Value of constr(state.pos) as 1D array.
+Value of constr(state.pos) as 1D array.
- Return type:
ArrayLike
@@ -1154,24 +1132,21 @@
-
dh2_flow_dmom(dt)
-Derivatives of h2_flow flow map with respect to input momentum.
+Derivatives of h2_flow flow map with respect to momentum argument.
- Parameters:
dt (ScalarLike) – Time interval flow simulated for.
- Returns:
-
-- Matrix representing derivative (Jacobian) of position output of
h2_flow with respect to the value of the momentum component of the
-initial input state.
-- dmom_dmom: 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:
-dpos_dmom
+-
@@ -1213,13 +1188,14 @@
-
grad_log_det_sqrt_gram(state)[source]
-Derivative of (half of) log-determinant of Gram matrix wrt position.
+Derivative of half of log-determinant of Gram matrix with respect to position
- Parameters:
state (ChainState) – State to compute value at.
- Returns:
-Value of log_det_sqrt_gram(state) derivative with respect to state.pos.
+Value of log_det_sqrt_gram(state) derivative with respect to
+state.pos.
- Return type:
ArrayLike
@@ -1252,8 +1228,9 @@
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
-where jacob_constr is the Jacobian of the constraint function constr and
-metric is the matrix representation of the metric on the ambient space.
+where jacob_constr is the Jacobian of the constraint function
+constr and metric is the matrix representation of the metric on
+the ambient space.
- Parameters:
state (ChainState) – State to compute value at.
@@ -1374,7 +1351,7 @@
state (ChainState) – State to compute value at.
- Returns:
-Value of Jacobian of constr(state.pos) as 2D array.
+Value of Jacobian of constr(state.pos) as 2D array.
- Return type:
ArrayLike
@@ -1386,9 +1363,9 @@
-
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.
+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:
@@ -1396,7 +1373,7 @@
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.
jacob_constr_1 if set to None.
- Return type:
@@ -1467,7 +1444,7 @@
- Returns:
-Projected momentum in the co-tangent space at state.pos.
+Projected momentum in the co-tangent space at state.pos.
- Return type:
ArrayLike
@@ -2152,7 +2129,7 @@
-
class mici.systems.EuclideanMetricSystem(neg_log_dens, metric=None, grad_neg_log_dens=None)[source]
-Bases: System
+Bases: TractableFlowSystem
Hamiltonian system with a Euclidean metric on the position space.
Here Euclidean metric is defined to mean a metric with a fixed positive definite
matrix representation \(M\). The momentum variables are taken to be independent
@@ -2249,24 +2226,21 @@
-
dh2_flow_dmom(dt)[source]
-Derivatives of h2_flow flow map with respect to input momentum.
+Derivatives of h2_flow flow map with respect to momentum argument.
- Parameters:
dt (ScalarLike) – Time interval flow simulated for.
- Returns:
-
-- Matrix representing derivative (Jacobian) of position output of
h2_flow with respect to the value of the momentum component of the
-initial input state.
-- dmom_dmom: 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:
-dpos_dmom
+-
@@ -2541,7 +2515,7 @@
state (ChainState) – State to compute value at.
- Returns:
-Value of constr(state.pos) as 1D array.
+Value of constr(state.pos) as 1D array.
- Return type:
ArrayLike
@@ -2603,24 +2577,21 @@
-
dh2_flow_dmom(dt)
-Derivatives of h2_flow flow map with respect to input momentum.
+Derivatives of h2_flow flow map with respect to momentum argument.
- Parameters:
dt (ScalarLike) – Time interval flow simulated for.
- Returns:
-
-- Matrix representing derivative (Jacobian) of position output of
h2_flow with respect to the value of the momentum component of the
-initial input state.
-- dmom_dmom: 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:
-dpos_dmom
+-
@@ -2662,13 +2633,14 @@
-
grad_log_det_sqrt_gram(state)
-Derivative of (half of) log-determinant of Gram matrix wrt position.
+Derivative of half of log-determinant of Gram matrix with respect to position
- Parameters:
state (ChainState) – State to compute value at.
- Returns:
-Value of log_det_sqrt_gram(state) derivative with respect to state.pos.
+Value of log_det_sqrt_gram(state) derivative with respect to
+state.pos.
- Return type:
ArrayLike
@@ -2701,8 +2673,9 @@
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
-where jacob_constr is the Jacobian of the constraint function constr and
-metric is the matrix representation of the metric on the ambient space.
+where jacob_constr is the Jacobian of the constraint function
+constr and metric is the matrix representation of the metric on
+the ambient space.
- Parameters:
state (ChainState) – State to compute value at.
@@ -2823,7 +2796,7 @@
state (ChainState) – State to compute value at.
- Returns:
-Value of Jacobian of constr(state.pos) as 2D array.
+Value of Jacobian of constr(state.pos) as 2D array.
- Return type:
ArrayLike
@@ -2835,9 +2808,9 @@
-
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.
+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:
@@ -2845,7 +2818,7 @@
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.
jacob_constr_1 if set to None.
- Return type:
@@ -2916,7 +2889,7 @@
- Returns:
-Projected momentum in the co-tangent space at state.pos.
+Projected momentum in the co-tangent space at state.pos.
- Return type:
ArrayLike
@@ -3059,24 +3032,21 @@
-
dh2_flow_dmom(dt)[source]
-Derivatives of h2_flow flow map with respect to input momentum.
+Derivatives of h2_flow flow map with respect to momentum argument.
- Parameters:
dt (ScalarLike) – Time interval flow simulated for.
- Returns:
-
-- Matrix representing derivative (Jacobian) of position output of
h2_flow with respect to the value of the momentum component of the
-initial input state.
-- dmom_dmom: 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:
-dpos_dmom
+-
@@ -4371,7 +4341,7 @@
depends only on the position variable however depending on the form of \(h_2\)
the corresponding exact Hamiltonian flow may or may not be simulable.
By default \(h_1\) is assumed to correspond to the negative logarithm of an
-unnormalized density on the position variables with respect to the Lebesgue measure,
+unnormalized density on the position variables 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.
@@ -4379,7 +4349,7 @@
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 the Lebesgue measure, with the corresponding
+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
@@ -4600,6 +4570,285 @@
+
+-
+class mici.systems.TractableFlowSystem(neg_log_dens, grad_neg_log_dens=None)[source]
+Bases: System
+Base class for Hamiltonian systems with tractable component flows.
+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
+Hamiltonian flows for both the \(h_1\) and \(h_2\) components are assumed to
+be tractable for subclasses of this class.
+By default \(h_1\) 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
+distribution it is wished to draw approximate samples from.
+
+- 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.
+
+
+-
+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(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)
+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)[source]
+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.
+
+
+
+
+-
+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 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
+
+
+
+
+
diff --git a/objects.inv b/objects.inv
index 0cd810d..1ee99b8 100644
Binary files a/objects.inv and b/objects.inv differ
diff --git a/searchindex.js b/searchindex.js
index c7cec5f..d941fdb 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"docnames": ["index", "mici", "mici.adapters", "mici.autodiff", "mici.autograd_wrapper", "mici.errors", "mici.integrators", "mici.interop", "mici.matrices", "mici.progressbars", "mici.samplers", "mici.solvers", "mici.stagers", "mici.states", "mici.systems", "mici.transitions", "mici.types", "mici.utils"], "filenames": ["index.rst", "mici.rst", "mici.adapters.rst", "mici.autodiff.rst", "mici.autograd_wrapper.rst", "mici.errors.rst", "mici.integrators.rst", "mici.interop.rst", "mici.matrices.rst", "mici.progressbars.rst", "mici.samplers.rst", "mici.solvers.rst", "mici.stagers.rst", "mici.states.rst", "mici.systems.rst", "mici.transitions.rst", "mici.types.rst", "mici.utils.rst"], "titles": ["Welcome to Mici\u2019s documentation!", "mici package", "mici.adapters module", "mici.autodiff module", "mici.autograd_wrapper module", "mici.errors module", "mici.integrators module", "mici.interop module", "mici.matrices module", "mici.progressbars module", "mici.samplers module", 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\ No newline at end of file
+Search.setIndex({"docnames": ["index", "mici", "mici.adapters", "mici.autodiff", "mici.autograd_wrapper", "mici.errors", "mici.integrators", "mici.interop", "mici.matrices", "mici.progressbars", "mici.samplers", "mici.solvers", "mici.stagers", "mici.states", "mici.systems", "mici.transitions", "mici.types", "mici.utils"], "filenames": ["index.rst", "mici.rst", "mici.adapters.rst", "mici.autodiff.rst", "mici.autograd_wrapper.rst", "mici.errors.rst", "mici.integrators.rst", "mici.interop.rst", "mici.matrices.rst", "mici.progressbars.rst", "mici.samplers.rst", "mici.solvers.rst", "mici.stagers.rst", "mici.states.rst", "mici.systems.rst", "mici.transitions.rst", "mici.types.rst", "mici.utils.rst"], "titles": ["Welcome to Mici\u2019s documentation!", "mici package", "mici.adapters module", "mici.autodiff module", "mici.autograd_wrapper module", "mici.errors module", "mici.integrators module", "mici.interop module", "mici.matrices module", "mici.progressbars module", "mici.samplers module", 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\ No newline at end of file