[Paper Review] Parameter elimination in particle Gibbs sampling

Anna Wigren, Riccardo Sven Risuleo+2 more|arXiv (Cornell University)|Oct 30, 2019

Statistical Methods and Bayesian Inference4 citations

TL;DR

This paper proposes marginalized particle Gibbs (mPG) and marginalized particle Gibbs with ancestor sampling (mPGAS) to improve Bayesian inference in state-space models by eliminating parameters from the state update step. By exploiting conjugate priors to marginalize parameters, the method achieves independent samples in the limit, reducing autocorrelation beyond what standard PG/PGAS can achieve, while maintaining linear-time complexity via sufficient statistics.

ABSTRACT

Bayesian inference in state-space models is challenging due to high-dimensional state trajectories. A viable approach is particle Markov chain Monte Carlo, combining MCMC and sequential Monte Carlo to form "exact approximations" to otherwise intractable MCMC methods. The performance of the approximation is limited to that of the exact method. We focus on particle Gibbs and particle Gibbs with ancestor sampling, improving their performance beyond that of the underlying Gibbs sampler (which they approximate) by marginalizing out one or more parameters. This is possible when the parameter prior is conjugate to the complete data likelihood. Marginalization yields a non-Markovian model for inference, but we show that, in contrast to the general case, this method still scales linearly in time. While marginalization can be cumbersome to implement, recent advances in probabilistic programming have enabled its automation. We demonstrate how the marginalized methods are viable as efficient inference backends in probabilistic programming, and demonstrate with examples in ecology and epidemiology.

Motivation & Objective

• To address the limitation of particle Gibbs and PGAS, which cannot outperform the intractable Gibbs sampler they approximate.
• To reduce autocorrelation in MCMC samples by eliminating parameters from the state update step.
• To enable efficient inference in state-space models with conjugate priors by transforming the model into a non-Markovian but linearly scalable form.
• To demonstrate the feasibility and efficiency of marginalized PMCMC methods in probabilistic programming frameworks like Birch.
• To provide a practical, automated approach to parameter marginalization using modern probabilistic programming tools.

Proposed method

• Marginalize out static parameters from the state update by exploiting conjugacy between prior and complete data likelihood.
• Derive marginalized versions of PG and PGAS (mPG/mPGAS) that sample states from the marginal posterior p(x₀:T|y₁:T), independent of parameters.
• Use sufficient statistics in the SMC algorithm to maintain linear-time complexity despite non-Markovian dependencies introduced by marginalization.
• Implement the method in the probabilistic programming language Birch, enabling automatic marginalization and integration with existing inference pipelines.
• Use blocking strategies to handle cases with diffuse priors, combining cSMC and mcSMC updates for improved robustness.
• Leverage recent advances in probabilistic programming to automate marginalization, reducing manual implementation burden.

Experimental results

Research questions

• RQ1 Can marginalizing parameters from the state update in particle Gibbs sampling lead to faster mixing and lower autocorrelation than standard PG/PGAS?
• RQ2 Does parameter marginalization enable the sampler to generate approximately independent draws in the large particle limit, even when the original Gibbs sampler produces dependent samples?
• RQ3 Can the computational cost of marginalized SMC remain linear in the number of timesteps despite the introduction of non-Markovian dependencies?
• RQ4 How can marginalization be automated and integrated into probabilistic programming frameworks for practical use?
• RQ5 What are the performance gains and limitations when applying marginalized PMCMC to real-world ecological and epidemiological models?

Key findings

• mPG and mPGAS achieve independent samples in the limit of infinite particles, as evidenced by the autocorrelation function dropping to zero at lag one, unlike standard PGAS which converges to the correlated Gibbs sampler.
• The marginalized samplers reduce autocorrelation significantly, with mPG showing faster convergence and lower dependence than PG and PGAS in both synthetic and real-world models.
• In the song sparrow population model, the estimated distribution of the density regulation parameter c matched literature values, validating the method’s accuracy.
• The method maintains linear-time complexity by using sufficient statistics, even though the resulting model is non-Markovian after marginalization.
• Numerical results show that mPG and mPGAS outperform standard PGAS in terms of effective sample size and convergence speed, particularly in high-dimensional state spaces.
• The implementation in Birch enables automatic marginalization, demonstrating that the method is practical and integrable into modern probabilistic programming workflows.