Distinguishing Hidden Markov Chains

TL;DR

This paper presents polynomial-time algorithms to determine whether two Hidden Markov Chains are distinguishable based on observation sequences, with methods that have exponentially decreasing error probabilities and applications in runtime verification.

Contribution

It introduces the first polynomial-time decision procedure for HMC distinguishability and develops two novel algorithms with different error properties.

Findings

Decidability of HMC distinguishability in polynomial time

Two algorithms with exponential error decay

Extension to multiple HMCs and application in runtime verification

Abstract

Hidden Markov Chains (HMCs) are commonly used mathematical models of probabilistic systems. They are employed in various fields such as speech recognition, signal processing, and biological sequence analysis. We consider the problem of distinguishing two given HMCs based on an observation sequence that one of the HMCs generates. More precisely, given two HMCs and an observation sequence, a distinguishing algorithm is expected to identify the HMC that generates the observation sequence. Two HMCs are called distinguishable if for every $\varepsilon > 0$ there is a distinguishing algorithm whose error probability is less than $\varepsilon$. We show that one can decide in polynomial time whether two HMCs are distinguishable. Further, we present and analyze two distinguishing algorithms for distinguishable HMCs. The first algorithm makes a decision after processing a fixed number of observations, and it exhibits two-sided error. The second algorithm processes an unbounded number of observations, but the algorithm has only one-sided error. The error probability, for both algorithms, decays exponentially with the number of processed observations. We also provide an algorithm for distinguishing multiple HMCs. Finally, we discuss an application in stochastic runtime verification.