# Efficient Computation of Updated Lower Expectations for Imprecise Continuous-Time Hidden Markov Chains

## TL;DR

This paper introduces a polynomial-time algorithm for efficiently computing lower expectations in imprecise continuous-time hidden Markov chains, accommodating various output variable types and model uncertainties.

## Contribution

It presents the first polynomial runtime algorithm for inference in imprecise continuous-time hidden Markov chains with flexible output variables.

## Key findings

- Algorithm achieves polynomial runtime for lower expectation computation.
- Supports both discrete and continuous output variables.
- Handles various types of model uncertainty using imprecise probabilities.

## Abstract

We consider the problem of performing inference with imprecise continuous-time hidden Markov chains, that is, imprecise continuous-time Markov chains that are augmented with random output variables whose distribution depends on the hidden state of the chain. The prefix `imprecise' refers to the fact that we do not consider a classical continuous-time Markov chain, but replace it with a robust extension that allows us to represent various types of model uncertainty, using the theory of imprecise probabilities. The inference problem amounts to computing lower expectations of functions on the state-space of the chain, given observations of the output variables. We develop and investigate this problem with very few assumptions on the output variables; in particular, they can be chosen to be either discrete or continuous random variables. Our main result is a polynomial runtime algorithm to compute the lower expectation of functions on the state-space at any given time-point, given a collection of observations of the output variables.