On the Computational Complexity of Hidden Markov Model Identification

2026-10-06

AI Summary

This study addresses the longstanding challenge that no complete algorithm exists for determining the identifiability of Hidden Markov Models (HMMs) and that its computational complexity remains unknown. From a computational complexity perspective, this work moves beyond traditional sufficient conditions to construct a complete algorithmic framework encompassing various notions of identifiability. Furthermore, it introduces a quantified reduction technique over the theory of reals to handle complex decision problems. The primary contributions are threefold: it establishes, for the first time, a PSPACE upper bound and a coETR lower bound for this problem; it demonstrates that multiple HMM identifiability problems can be decided within PSPACE; and it precisely characterizes the computational complexity boundaries of HMM identifiability determination.

Abstract

Identification is the task of recovering the parameters of an unknown ground-truth model from sampled data. When parameters other than the ground truth induce the same output distribution, data alone does not provide enough information to recover the ground truth, and the model is thus called unidentifiable. We study the identifiability problem for hidden Markov models (HMMs): given an HMM, is it identifiable? Existing work on HMM identification establishes conditions under which the ground-truth HMM can be identified. However, most of these conditions are sufficient but not necessary, meaning that, when a model does not satisfy them, its identifiability remains inconclusive. We instead take a computational perspective: is there a sound and complete algorithm that decides whether a given HMM is identifiable, and if so, what is the complexity of this decision problem? We consider the decision problems arising from the various notions of identifiability in the literature, including deterministic, generic, global, local, state-permutation- invariant, and finite-alphabet identifiability. We show that all of these problems are decidable in PSPACE, via reductions to the theory of the reals at various levels of its quantifier-alternation hierarchy. We further show that the deterministic variants are already coETR-hard (and hence coNP-hard) for simply parameterized families.

Problem

• Hidden Markov Model
• Identifiability
• Computational Complexity
• Decision Problem

Innovation

• Hidden Markov Model
• Identifiability
• Computational Complexity
• PSPACE
• Theory of the Reals