# Manifold MCMC methods for inference in diffusion models

Code accompanying the paper Manifold Markov chain Monte Carlo methods for Bayesian inference in a wide class of diffusion models.

The manifold Markov chain Monte Carlo (MCMC) methods in Mici are used for inference, with the sde package in this repository adding helper functions and classes specific to performing inference in stochastic differential equation (SDE) models, in particular diffusion processes. The numerical integrators for SDE systems in sde.integrators and example models in sde.example_models use SymNum to automatically construct derivatives of model specific functions. JAX is used to generate efficient just-in-time compiled functions for numerically integrating the diffusion models and evaluating the derivatives required for performing MCMC inference.

Installation

The sde package requires Python 3.11 or above. To install the sde package and its dependencies in the current Python environment run

bash pip install git+https://github.com/thiery-lab/manifold-mcmc-for-diffusions.git

To install the package and all dependencies required to run the scripts for reproducing the experiments in the paper run

bash pip install git+https://github.com/thiery-lab/manifold-mcmc-for-diffusions.git#egg=sde\[scripts\]

To install the package and all dependencies required to run the example Jupyter notebook run

bash pip install git+https://github.com/thiery-lab/manifold-mcmc-for-diffusions.git#egg=sde\[notebook\]

Experiment scripts

A number of scripts for reproducing the numerical experiments used to produce the figures in the paper are provided in the scripts directory. To run these scripts the sde package, its dependencies and the additional dependencies required to run the scripts need to be installed in the local Python environment as described above.

For the experiments with the FitzHugh–Nagumo model with noisy observations, the MCMC algorithm implemented in the Julia BridgeSDEInference package is also used for comparison. A local Julia installation (tested with version 1.6) is therefore also required to run these experiments. The required Julia dependencies can be installed from a Julia REPL by running

Julia import Pkg Pkg.add(["ArgParse", "BridgeSDEInference", "JSON", "NPZ", "PyCall"])

There are Python scripts for running individual experiments with each model and inference algorithm combination, as well as Python scripts for generating plots from the results for each model. A description of what a particular script does, what arguments can be passed and what their default values are can be displayed by running

bash python path/to/script.py --help

There are also bash scripts provided for running the full set of experiments used to produce the figures in the paper. As a warning, these scripts sequentially run experiments over grids of different variables and in some cases for multiple different pseudo-random seeds, and so running all these scripts should be expected to take several days to complete. As some of the plotted values are dependent on computation times which is affected among other things by the system being run on, the generated plots will not exactly match those in the paper.

Example notebook

A complete example of applying the method described in the paper to perform inference in a Fitzhugh-Nagumo hypoelliptic diffusion model with accompanying explanatory notes is provided in the Jupyter notebook FitzHugh-Nagumo_example.ipynb . This notebook can also be viewed or run interactively online using the links below.

FitzHugh-Nagumo_example.ipynb
Open non-interactive version with nbviewer
Open interactive version with Binder
Open interactive version with Google Colab

Citation

To cite the pre-print the following bibtex entry can be used

bibtex @article{graham2019manifold, author={Graham, Matthew M. and Thiery, Alexandre H. and Beskos, Alexandros}, title={Manifold Markov chain Monte Carlo methods for Bayesian inference in a wide class of diffusion models}, year={2019}, journal={Pre-print arxiv:1912.02982}, url={https://arxiv.org/abs/1912.02982} }