An efficient algorithm for estimating state sequences in imprecise hidden Markov models

TL;DR

This paper introduces an efficient exact algorithm for estimating state sequences in imprecise hidden Markov models, which handle uncertainty using coherent lower previsions, improving robustness over traditional precise models.

Contribution

The paper presents a novel algorithm for exact state sequence estimation in imprecise HMMs, extending traditional methods to models with uncertainty represented by coherent lower previsions.

Findings

Algorithm has quadratic complexity in sequence length

Number of maximal sequences varies with model parameters

Demonstrated robustness in optical character recognition

Abstract

We present an efficient exact algorithm for estimating state sequences from outputs (or observations) in imprecise hidden Markov models (iHMM), where both the uncertainty linking one state to the next, and that linking a state to its output, are represented using coherent lower previsions. The notion of independence we associate with the credal network representing the iHMM is that of epistemic irrelevance. We consider as best estimates for state sequences the (Walley--Sen) maximal sequences for the posterior joint state model conditioned on the observed output sequence, associated with a gain function that is the indicator of the state sequence. This corresponds to (and generalises) finding the state sequence with the highest posterior probability in HMMs with precise transition and output probabilities (pHMMs). We argue that the computational complexity is at worst quadratic in the length of the Markov chain, cubic in the number of states, and essentially linear in the number of maximal state sequences. For binary iHMMs, we investigate experimentally how the number of maximal state sequences depends on the model parameters. We also present a simple toy application in optical character recognition, demonstrating that our algorithm can be used to robustify the inferences made by precise probability models.