# Fundamental limits for learning hidden Markov model parameters

arXiv: https://arxiv.org/abs/2106.12936 · HTML: https://mlsanity.com/paper/2106.12936/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 48 references that are in the map.
- First posted on arXiv 24 Jun 2021.

## Builds on (most cited first, 4 of 4)

- [Statistical clustering of temporal networks through a dynamic stochastic block model](https://mlsanity.com/paper/1506.07464/) (2015)
- [Optimal estimation of high-dimensional location Gaussian mixtures](https://mlsanity.com/paper/2002.05818/) (2020)
- [Gene Hunting with Knockoffs for Hidden Markov Models](https://mlsanity.com/paper/1706.04677/) (2017)
- [Joint self-supervised blind denoising and noise estimation](https://mlsanity.com/paper/2102.08023/) (2021)

## Built on by (most cited first, 2 of 2)

- [Adaptive Mean Estimation in the Hidden Markov sub-Gaussian Mixture Model](https://mlsanity.com/paper/2406.12446/) (2024)
- [Nous: Learning and Certifying Memory Decisions Before Source Calibration](https://mlsanity.com/paper/2610.00094/) (2026)

## Abstract (arXiv)

We study the frontier between learnable and unlearnable hidden Markov models (HMMs). HMMs are flexible tools for clustering dependent data coming from unknown populations. The model parameters are known to be fully identifiable (up to label-switching) without any modeling assumption on the distributions of the populations as soon as the clusters are distinct and the hidden chain is ergodic with a full rank transition matrix. In the limit as any one of these conditions fails, it becomes impossible in general to identify parameters. For a chain with two hidden states we prove nonasymptotic minimax upper and lower bounds, matching up to constants, which exhibit thresholds at which the parameters become learnable. We also provide an upper bound on the relative entropy rate for parameters in a neighbourhood of the unlearnable region which may have interest in itself.