The Statistical Geometry of Latent Recovery in Hidden Markov Models
Abstract
Hidden Markov models (HMMs) offer an important capability to interpret fitted latent states and transitions. When emission distributions nearly coincide, a small observation-density error does not ensure accurate estimation of the latent quantities needed for interpretation. This motivates us to characterize convergence rates for latent recovery that account for emission separation. Our main contribution lies in deriving deterministic inverse inequalities that convert Hellinger error between fitted and true densities into bounds on latent recovery errors. We develop these bounds for multivariate Gaussian HMMs with a common, known emission covariance matrix and combine them with a Hellinger estimation guarantee for the global maximum likelihood estimator to obtain latent recovery rates. Although the joint observation distribution is a Gaussian mixture with exponentially many path components, they share the same emission centers across time. Leveraging this shared structure, we show that the separation exponents are governed by the maximum number of simultaneously approaching centers rather than the number of path components. Under exact specification, the global bounds on state-sequence probability discrepancies and the 1-Wasserstein loss for latent location distributions exhibit sharp separation exponents that do not increase with the number of selected times. Moreover, sufficiently small observation-density error guarantees unique matching between fitted and true emission centers, which yields error bounds for individual emission centers and transition rows. Under over-specification, individual split-state transition rows are not necessarily identifiable, so we quantify aggregate latent recovery errors and show that state-sequence probability bounds incur larger separation exponents compared to exact specification. We conduct synthetic experiments to validate our theoretical results.
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BibTeX
@misc{pham2026statistical,
title = {{The Statistical Geometry of Latent Recovery in Hidden Markov Models}},
author = {Trang Pham and Dung Le and Alessandro Rinaldo and Nhat Ho},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00680},
eprint = {2611.00680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00680},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Statistical Geometry of Latent Recovery in Hidden Markov Models
AU - Trang Pham
AU - Dung Le
AU - Alessandro Rinaldo
AU - Nhat Ho
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00680
UR - https://arxiv.org/abs/2611.00680
AB - Hidden Markov models (HMMs) offer an important capability to interpret fitted latent states and transitions. When emission distributions nearly coincide, a small observation-density error does not ensure accurate estimation of the latent quantities needed for interpretation. This motivates us to characterize convergence rates for latent recovery that account for emission separation. Our main contribution lies in deriving deterministic inverse inequalities that convert Hellinger error between fitted and true densities into bounds on latent recovery errors. We develop these bounds for multivariate Gaussian HMMs with a common, known emission covariance matrix and combine them with a Hellinger estimation guarantee for the global maximum likelihood estimator to obtain latent recovery rates. Although the joint observation distribution is a Gaussian mixture with exponentially many path components, they share the same emission centers across time. Leveraging this shared structure, we show that the separation exponents are governed by the maximum number of simultaneously approaching centers rather than the number of path components. Under exact specification, the global bounds on state-sequence probability discrepancies and the 1-Wasserstein loss for latent location distributions exhibit sharp separation exponents that do not increase with the number of selected times. Moreover, sufficiently small observation-density error guarantees unique matching between fitted and true emission centers, which yields error bounds for individual emission centers and transition rows. Under over-specification, individual split-state transition rows are not necessarily identifiable, so we quantify aggregate latent recovery errors and show that state-sequence probability bounds incur larger separation exponents compared to exact specification. We conduct synthetic experiments to validate our theoretical results.
ER -