Efficient Bayesian Inference for Discretely Observed Continuous Time Markov Chains

TL;DR

This paper introduces a scalable Bayesian method for inferring continuous-time Markov chains from discrete observations, using a pseudo-likelihood approach that remains computationally efficient in higher dimensions.

Contribution

It proposes a novel pseudo-likelihood framework that estimates the transition matrix and spectral decomposition jointly, with theoretical guarantees and scalable Gibbs sampling.

Findings

Computational cost is nearly invariant to data size and scales well with dimension.

Theoretical guarantees include Bernstein-von Mises theorem and posterior consistency.

Method demonstrates robustness and flexibility in simulations and real applications.

Abstract

Inference for continuous-time Markov chains (CTMCs) becomes challenging when the process is only observed at discrete time points. The exact likelihood is intractable, and existing methods often struggle even in medium-dimensional state-spaces. We propose a scalable Bayesian framework for CTMC inference based on a pseudo-likelihood that bypasses the need for the full intractable likelihood. Our approach jointly estimates the probability transition matrix and a biorthogonal spectral decomposition of the generator, enabling an efficient Gibbs sampling procedure that obeys embeddability. Existing methods typically integrate out the unobserved transitions, which becomes computationally burdensome as the number of data or dimensions increase. The computational cost of our method is near-invariant in the number of data and scales well to medium-high dimensions. We justify our pseudo-likelihood approach by establishing theoretical guarantees, including a Bernstein-von Mises theorem for the probability transition matrix and posterior consistency for the spectral parameters of the generator. Through simulation and applications, we showcase the flexibility and robustness of our approach, offering a tractable and scalable approach to Bayesian inference for CTMCs.