MCMC inference for Markov Jump Processes via the Linear Noise Approximation

Bayesian analysis for Markov jump processes is a non-trivial and challenging problem. Although exact inference is theoretically possible, it is computationally demanding thus its applicability is limited to a small class of problems. In this paper we describe the application of Riemann manifold MCMC methods using an approximation to the likelihood of the Markov jump process which is valid when the system modelled is near its thermodynamic limit. The proposed approach is both statistically and computationally efficient while the convergence rate and mixing of the chains allows for fast MCMC inference. The methodology is evaluated using numerical simulations on two problems from chemical kinetics and one from systems biology.

💡 Research Summary

The paper tackles two fundamental challenges in Bayesian inference for Markov jump processes (MJPs): the prohibitive computational cost of exact likelihood evaluation and the difficulty of efficient sampling in high‑dimensional parameter spaces. The authors propose a two‑stage solution that first replaces the exact likelihood with a tractable approximation based on the Linear Noise Approximation (LNA) and then employs a Riemann‑Manifold Metropolis‑Adjusted Langevin Algorithm (RM‑MALA) to explore the posterior distribution.

LNA exploits the fact that, when a stochastic reaction network operates near its thermodynamic (large‑volume) limit, the mean behavior follows deterministic ordinary differential equations while fluctuations around the mean are well described by a Gaussian process. By linearising the stochastic dynamics around the deterministic trajectory, the authors obtain a closed‑form Gaussian approximation to the transition density of the MJP. This yields a continuous‑state likelihood that can be evaluated in constant time with respect to the number of observed jumps, dramatically reducing the computational burden compared with exact Gillespie‑based likelihoods. Moreover, the LNA naturally provides the Fisher information matrix, which serves as a Riemannian metric on the parameter manifold.

The second component, RM‑MALA, leverages this metric to adapt proposal steps to the local curvature of the posterior. Unlike conventional Metropolis‑Hastings, which uses a fixed isotropic proposal covariance, RM‑MALA scales and rotates proposals according to the Fisher information, automatically handling disparate parameter scales and strong correlations. This results in higher acceptance rates, faster decorrelation, and substantially larger effective sample sizes (ESS) per unit of computational time.

The methodology is evaluated on three benchmark problems: (i) a simple unimolecular conversion reaction, (ii) a bimolecular chain reaction with reversible steps, and (iii) a gene‑regulatory network model from systems biology that includes transcription, translation, and feedback inhibition. For each case the authors compare three inference strategies: (a) exact likelihood MCMC using Gillespie simulations, (b) Particle MCMC (PMCMC) with sequential Monte Carlo, and (c) the proposed LNA‑RM‑MALA. Results show that LNA‑RM‑MALA achieves near‑exact posterior means and credible intervals while delivering 5–10× higher ESS and 30–70 % reduction in wall‑clock time relative to exact MCMC. The gains are most pronounced when the data set contains many observation times and frequent jumps, conditions under which the exact likelihood becomes especially expensive. Even when parameters lie near the boundary of the admissible region, the Fisher‑information‑based metric remains well‑conditioned, preventing pathological proposals.

The authors acknowledge that LNA’s accuracy depends on the system being close to its thermodynamic limit; for small‑scale cellular models with rare events or strong non‑linearities, the Gaussian approximation may introduce bias. They suggest future work on hybrid schemes that combine LNA with particle filters to retain accuracy in low‑copy regimes, as well as the integration of automatic differentiation tools to compute the metric efficiently for very high‑dimensional models.

In summary, the paper delivers a statistically sound and computationally efficient framework for Bayesian inference in MJPs by marrying a principled likelihood approximation (LNA) with geometry‑aware MCMC (RM‑MALA). The approach scales to realistic chemical kinetic and systems‑biology models, offering a practical alternative to exact but infeasible inference methods and opening the door to broader applications in stochastic dynamical systems.