Pseudo-marginal Inference for CTMCs on Infinite Spaces via Monotonic Likelihood Approximations

TL;DR

This paper introduces a new method for Bayesian inference on infinite-state CTMCs that uses monotonic likelihood approximations to improve efficiency without computing exact matrix exponentials.

Contribution

The authors develop a novel unbiased likelihood estimation technique using doubly-monotone matrix exponential approximations, including a new skeletoid approximation, enhancing inference efficiency.

Findings

The approach significantly improves inference efficiency for various CTMCs.

The skeletoid approximation reduces variance in likelihood estimates.

The method maintains unbiasedness while avoiding costly matrix exponential computations.

Abstract

Bayesian inference for Continuous-Time Markov Chains (CTMCs) on countably infinite spaces is notoriously difficult because evaluating the likelihood exactly is intractable. One way to address this challenge is to first build a non-negative and unbiased estimate of the likelihood -- involving the matrix exponential of finite truncations of the true rate matrix -- and then to use the estimates in a pseudo-marginal inference method. In this work, we show that we can dramatically increase the efficiency of this approach by avoiding the computation of exact matrix exponentials. In particular, we develop a general methodology for constructing an unbiased, non-negative estimate of the likelihood using doubly-monotone matrix exponential approximations. We further develop a novel approximation in this family -- the skeletoid -- as well as theory regarding its approximation error and how that relates to the variance of the estimates used in pseudo-marginal inference. Experimental results show that our approach yields more efficient posterior inference for a wide variety of CTMCs.