…sion

`fit_pathfinder`
- Edited `fit_pathfinder` to produce `pathfinder_state`, `pathfinder_info`, `pathfinder_samples` and `pathfinder_idata` for closer examination of the outputs.
- Changed the `num_samples` argument name to `num_draws` to avoid `TypeError` got multiple values for keyword argument 'num_samples'.
- Initial points are automatically set to jitter as jitter is required for pathfinder.

Extras
- New function 'get_jaxified_logp_ravel_inputs' to simplify previous code structure in fit_pathfinder.

Tests
- Added extra test for pathfinder to test pathfinder_info variables and pathfinder_idata  are consistent for a given random seed.

Add a new PyMC-based implementation of Pathfinder VI that uses PyTensor operations which provides support for both PyMC and BlackJAX backends in fit_pathfinder.

- Implemented  in  to support running multiple Pathfinder instances in parallel.
- Implemented  function in  for Pareto Smoothed Importance Resampling (PSIR).
- Moved relevant pathfinder files into the  directory.
- Updated tests to reflect changes in the Pathfinder implementation and added tests for new functionalities.

…nd .

…race data to InferenceData

… for bfgs_sample

Summaryh of changes:
- Remove multiprocessing code in favour of reusing compiled  for each path
-  takes only random_seed as argument for each path
- Compute graph significantly smaller by using pure pytensor op and symoblic variables
- Added LBFGSOp to compile with pytensor.function
- Cleaned up codes using pytensor variables

…and .

- Corrected the dimensions in comments for matrices Q and R in the  function.
- Uumerical stability in the  calculation by changing from  to .

Fixed incorrect and inconsistent posterior approximations in the Pathfinder VI
algorithm by:

1. Adding missing parentheses in the phi calculation to ensure proper order
   of operations in matrix multiplications
2. Changing the sign in mu calculation from 'x +' to 'x -' to match Stan's
   implementation (which differs from the original paper)

The resulting changes now make the posterior approximations more reliable.

Implements both sparse and dense BFGS sampling approaches for Pathfinder VI:
- Adds bfgs_sample_dense for cases where 2*maxcor >= num_params.
- Moved existing  and  computations to bfgs_sample_sparse, making the sparse use cases more explicit.

Other changes:
- Sets default maxcor=5 instead of dynamic sizing based on parameters

Dense approximations are recommended when the target distribution has higher dependencies among the parameters.

Bigger changes:
- Made pmx.fit compatible with method='pathfinder'
- Remove JAX dependency when inference_backend='pymc' to support Windows users
- Improve runtime performance by setting trust_input=True for compiled functions

Minor changes:
- Change default num_paths from 1 to 4 for stable and reliable approximations
- Change LBFGS code using dataclasses
- Update tests to handle both PyMC and BlackJAX backends

- Add LBFGSInitFailed exception for failed LBFGS initialisation
- Skip failed paths in multipath_pathfinder and track number of failures
- Handle NaN values from Cholesky decompsition in bfgs_sample
- Add checks for numericl stabilty in matrix operations

Slight performance improvements:
- Set allow_gc=False in scan ops
- Use FAST_RUN mode consistently

Major:
  - Added progress bar support.

Minor
  - Added  exception for non-finite log prob values
  - Removed .
  - Allowed maxcor argument to be None, and dynamically set based on the number of model parameters.
  - Improved logging to inform users about failed paths and lbfgs initialisation.