Forward-Backward Latent State Inference for Hidden Continuous-Time semi-Markov Chains
Nicolai Engelmann,Heinz Koeppl
Hidden semi-Markov Models (HSMMs) - 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 HSMMs can be generalized to latent Continuous-Time semi-Markov Chains (CTSMCs). 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 HSMMs are introduced. We evaluate our approaches in latent state inference scenarios in comparison to classical HSMMs.
