  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.
