On Computing the Total Variation Distance of Hidden Markov Models

TL;DR

This paper investigates the computational complexity of calculating the total variation distance between hidden Markov models, revealing undecidability and hardness results that highlight fundamental limits in model comparison.

Contribution

It proves undecidability of the exact distance and establishes #P-hardness and PSPACE membership for approximation, advancing understanding of computational boundaries in HMM analysis.

Findings

Exact distance computation is undecidable.

Approximation is #P-hard.

Distance approximation lies in PSPACE.

Abstract

We prove results on the decidability and complexity of computing the total variation distance (equivalently, the $L_1$-distance) of hidden Markov models (equivalently, labelled Markov chains). This distance measures the difference between the distributions on words that two hidden Markov models induce. The main results are: (1) it is undecidable whether the distance is greater than a given threshold; (2) approximation is #P-hard and in PSPACE.