[Paper Review] Scalable Bayesian Inference for Population Markov Jump Processes

Iker Perez, Theodore Kypraios|arXiv (Cornell University)|Apr 17, 2019

Gene Regulatory Network AnalysisBiochemistry, Genetics and Molecular Biology26 references3 citations

TL;DR

This paper introduces a scalable, general-purpose data-augmentation framework for exact Bayesian inference in population Markov jump processes using uniformized representations of non-stationary jump processes. It enables efficient MCMC sampling across diverse models—epidemics, immigration, predator-prey—by leveraging auxiliary variables that improve mixing and scalability, especially in large or infinite state-space systems.

ABSTRACT

Bayesian inference for Markov jump processes (MJPs) where available observations relate to either system states or jumps typically relies on data-augmentation Markov Chain Monte Carlo. State-of-the-art developments involve representing MJP paths with auxiliary candidate jump times that are later thinned. However, these algorithms are i) unfeasible in situations involving large or infinite capacity systems and ii) not amenable for all observation types. In this paper we establish and present a general data-augmentation framework for population MJPs based on uniformized representations of the underlying non-stationary jump processes. This leads to multiple novel MCMC samplers which enable exact (in the Monte Carlo sense) inference tasks for model parameters. We show that proposed samplers outperform existing popular approaches, and offer substantial efficiency gains in applications to partially observed stochastic epidemics, immigration processes and predator-prey dynamical systems.

Motivation & Objective

• Address the limitations of existing MCMC methods for population Markov jump processes, which are computationally infeasible for large or infinite state spaces.
• Overcome the inapplicability of current data-augmentation schemes to diverse observation types, such as population counts or jump events.
• Develop a general-purpose framework that supports exact Bayesian inference (in the Monte Carlo sense) for non-stationary, time-inhomogeneous jump processes.
• Enable scalable inference in complex systems such as stochastic epidemics, immigration-death processes, and predator-prey dynamics with partial or noisy observations.
• Design efficient MCMC samplers that outperform existing state-of-the-art methods in terms of effective sample size and computational efficiency.

Proposed method

• Utilizes uniformized representations of non-stationary jump processes to construct a general data-augmentation framework for MJP inference.
• Introduces auxiliary variables based on Poisson thinning of uniformized jump processes, enabling exact path augmentation without path rejection.
• Employs a dominating rate function $\Omega = \max_{x \in \mathcal{S}} \sup_{t \in [0,T]} |Q_x(t)|$ to control the uniformization rate, ensuring valid stochastic simulation.
• Derives novel MCMC samplers that jointly update latent jump times and paths using conditional densities derived from the uniformized path density in Equation (1).
• Applies operator-based thinning strategies (e.g., $\psi(t,x) = 0.5 \cdot |Q_x(t)|$) to improve mixing and reduce autocorrelation in MCMC chains.
• Implements lag-based conditioning bridges (e.g., $l = 0.5N$ or $0.75N$) to enhance mixing in high-dimensional state spaces, particularly in large-population systems.

Experimental results

Research questions

• RQ1 Can a general-purpose data-augmentation framework be developed for exact Bayesian inference in non-stationary, population-scale Markov jump processes?
• RQ2 How can auxiliary-variable MCMC methods be designed to remain efficient and scalable in systems with large or infinite state spaces?
• RQ3 To what extent do the proposed samplers outperform existing state-of-the-art methods in terms of effective sample size and computational efficiency?
• RQ4 Can the framework handle diverse observation types, including population counts and jump events, across different stochastic models?
• RQ5 What is the impact of lag-based conditioning and adaptive thinning operators on MCMC mixing and convergence in high-dimensional inference tasks?

Key findings

• The proposed framework enables exact Bayesian inference for population Markov jump processes using MCMC samplers based on uniformized path representations.
• The new MCMC samplers achieve substantial efficiency gains, with effective sample size ratios exceeding 1.5× compared to vanilla uniformization and existing benchmarks in large-population settings.
• In predator-prey models with state space up to $14,400$ (i.e., $120^2$), the method scales effectively, maintaining high effective sample sizes even at high population bounds.
• Lag-based conditioning bridges with $l = 0.5N$ and $l = 0.75N$ significantly improve mixing, outperforming both plain and reference auxiliary-variable samplers.
• The framework subsumes and generalizes prior methods (e.g., Rao and Teh, 2013) as special cases, confirming theoretical consistency.
• Empirical results show that the proposed samplers maintain high performance across all parameters ($\alpha, \beta, \delta, \gamma$), with minimum effective sample size ratios consistently above 1.2 across all tested configurations.