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

arXiv: https://arxiv.org/abs/2210.09058 · HTML: https://mlsanity.com/paper/2210.09058/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 17 Oct 2022.

## Builds on (most cited first, 4 of 4)

- [Model-based Reinforcement Learning for Semi-Markov Decision Processes with Neural ODEs](https://mlsanity.com/paper/2006.16210/) (2020)
- [Learning from Clinical Judgments: Semi-Markov-Modulated Marked Hawkes Processes for Risk Prognosis](https://mlsanity.com/paper/1705.05267/) (2017)
- [A Hidden Absorbing Semi-Markov Model for Informatively Censored Temporal Data: Learning and Inference](https://mlsanity.com/paper/1612.06007/) (2016)
- [Mathematical understanding of detailed balance condition violation and its application to Langevin dynamics](https://mlsanity.com/paper/1503.02356/) (2015)

## Abstract (arXiv)

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.