Gradient-based conditional SMC for inference of high-dimensional Markovian systems

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This repository contains the code for the paper Particle-MALA and Particle-mGRAD: Gradient-based MCMC methods for high-dimensional state-space models by Adrien Corenflos and Axel Finke, see also the attached preprint. It contains both a general implementation of the algorithms considered and the code to reproduce the experiments.

Abstract

State-of-the-art methods for Bayesian inference in state-space models are (a) conditional sequential Monte Carlo (CSMC) algorithms; (b) sophisticated ‘classical’ MCMC algorithms like MALA, or mGRAD from Titsias and Papaspiliopoulos (2018). The former propose $N$ particles at each time step to exploit the model's ‘decorrelation-over-time’ property and thus scale favourably with the time horizon, $T$, but break down if the dimension of the latent states, $D$, is large. The latter leverage gradient- or prior-informed local proposals to scale favourably with $D$ but exhibit sub-optimal scalability with $T$ due to a lack of model-structure exploitation.

We introduce methods which combine the strengths of both approaches.

The first, Particle-MALA, spreads $N$ particles locally around the current state using gradient information, thus extending MALA to $T > 1$ time steps and $N > 1$ proposals. The second, Particle-mGRAD, additionally incorporates (conditionally) Gaussian prior dynamics into the proposal, thus extending the mGRAD algorithm to $T > 1$ time steps and $N > 1$ proposals. We prove that Particle-mGRAD interpolates between CSMC and Particle-MALA, resolving the ‘tuning problem’ of choosing between CSMC (superior for highly informative prior dynamics) and Particle-MALA (superior for weakly informative prior dynamics).

We similarly extend other ‘classical’ MCMC approaches like auxiliary MALA, aGRAD, and preconditioned Crank–Nicolson–Langevin (PCNL) to $T > 1$ time steps and $N > 1$ proposals. In experiments, for both highly and weakly informative prior dynamics, our methods substantially improve upon both CSMC and sophisticated ‘classical’ MCMC approaches.

Expected behaviour

Scaling of the methods with the dimension	Scaling of the methods with the number of time steps	Scaling for informative vs. uninformative dynamics

Installation

Create a clean environment using your favourite environment manager. Run pip install . (or if you use poetry follow their instructions) in the root folder of the repository.

Usage

The following methods are given in order of decreasing generality: 1. applies to all the models, 2. does not apply to 1., 3. does not apply to 1. and 2., etc.

1. The system is simply defined as a product of bi-variate potential functions. This is the most general case, the user can use one of the following methods:
   • rw_csmc.py for the Particle-RWM algorithm.
   • al_csmc_f.py (auxiliary Langevin CSMC with filtering gradients) named Particle-aMALA in the paper
   • al_csmc_s.py (auxiliary Langevin CSMC with smoothing gradients) named Particle-aMALA+ in the paper
   • l_csmc_f.py (marginal MALA CSMC with filtering gradients) named Particle-MALA in the paper
2. The system is given as a generic Feynman–Kac model. In this case, should simply use the csmc.py file.
3. The system is given as a Feynman–Kac model with conditionally Gaussian dynamics. In this case, the user can use
   • atp_csmc_f.py (auxiliary Titsias–Papaspiliopoulos CSMC with filtering gradients) named Particle-aGRAD in the paper
   • atp_csmc_s.py (auxiliary Titsias–Papaspiliopoulos CSMC with smoothing gradients) named Particle-aGRAD+ in the paper. Note that this recovers Particle-aGRAD if the potentials are univariate.
   • tp_csmc_f.py (marginal Titsias–Papaspiliopoulos CSMC with filtering gradients) named Particle-mGRAD in the paper
4. The system is given as a Feynman–Kac model with fully Gaussian dynamics. In this case, the user can use
   • t_atp_csmc_f.py (twisted auxiliary Titsias–Papaspiliopoulos CSMC with filtering gradients) named twisted Particle-aGRAD in the paper

Additionally to these, a MALA algorithm (potentially with multiple proposals) is given in mala.py and an aGRAD algorithm (potentially with multiple proposals) is given in grad.py. For a description of the algorithms, we refer to our article, and for a description of the arguments of each method, to the corresponding file.

Reproducing the experiments

Details are given in the README file in the corresponding folder.