Forward-Backward Latent State Inference for Hidden Continuous-Time semi-Markov Chains

TL;DR

This paper extends latent state inference methods from discrete-time HSMMs to continuous-time semi-Markov chains, enabling more accurate modeling of irregularly spaced event data with scalable algorithms.

Contribution

It introduces a generalized inference framework for continuous-time semi-Markov chains, including integro-differential equations and a scalable Viterbi algorithm, improving modeling of irregular event data.

Findings

Efficient numerical solutions for the new equations.

Introduction of variable-step HSMMs for practical use.

Improved inference accuracy over classical HSMMs.

Abstract

Hidden semi-Markov Models (HSMM's) - while broadly in use - are restricted to a discrete and uniform time grid. They are thus not well suited to explain often irregularly spaced discrete event data from continuous-time phenomena. We show that non-sampling-based latent state inference used in HSMM's can be generalized to latent Continuous-Time semi-Markov Chains (CTSMC's). We formulate integro-differential forward and backward equations adjusted to the observation likelihood and introduce an exact integral equation for the Bayesian posterior marginals and a scalable Viterbi-type algorithm for posterior path estimates. The presented equations can be efficiently solved using well-known numerical methods. As a practical tool, variable-step HSMM's are introduced. We evaluate our approaches in latent state inference scenarios in comparison to classical HSMM's.