Asymptotics and finite sample bounds for prediction and smoothing in Wright-Fisher hidden Markov models
Abstract
We study signal prediction and smoothing in hidden Markov models whose latent signal is a multi-type Wright-Fisher diffusion and whose discrete-time observations are categorical. We consider a repeated-sampling regime with finitely many inspection times and large within-time samples. Under neutrality and parent-independent mutation, the Bayesian predictive and smoothing laws admit exact representations as mixtures of Dirichlet distributions. We show that the one-step predictive and local smoothing laws converge in total variation to the corresponding Wright-Fisher transition and bridge laws determined by the neighboring population frequencies, thereby justifying plug-in approximations based on empirical proportions. For prediction, we derive explicit finite-sample bounds for this approximation and for bounded predictive functionals such as diversity, together with a finite-sample concentration bound for the marginal smoother. The joint posterior at the inspection times concentrates at the target frequencies and satisfies a Bernstein-von Mises limit with distinct interior and boundary regimes, with Gaussian and Gamma limits on different scales. The results allow arbitrary dependence across inspection times. A key technical ingredient is a fixed-interval tail bound for the coalescent-based dual death process.
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BibTeX
@misc{malgieri2026asymptotics,
title = {{Asymptotics and finite sample bounds for prediction and smoothing in Wright-Fisher hidden Markov models}},
author = {Luigi M. Malgieri and Filippo Ascolani and Matteo Giordano and Matteo Ruggiero},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22444},
eprint = {2609.22444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22444},
}
RIS
TY - UNPB
TI - Asymptotics and finite sample bounds for prediction and smoothing in Wright-Fisher hidden Markov models
AU - Luigi M. Malgieri
AU - Filippo Ascolani
AU - Matteo Giordano
AU - Matteo Ruggiero
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2609.22444
UR - https://arxiv.org/abs/2609.22444
AB - We study signal prediction and smoothing in hidden Markov models whose latent signal is a multi-type Wright-Fisher diffusion and whose discrete-time observations are categorical. We consider a repeated-sampling regime with finitely many inspection times and large within-time samples. Under neutrality and parent-independent mutation, the Bayesian predictive and smoothing laws admit exact representations as mixtures of Dirichlet distributions. We show that the one-step predictive and local smoothing laws converge in total variation to the corresponding Wright-Fisher transition and bridge laws determined by the neighboring population frequencies, thereby justifying plug-in approximations based on empirical proportions. For prediction, we derive explicit finite-sample bounds for this approximation and for bounded predictive functionals such as diversity, together with a finite-sample concentration bound for the marginal smoother. The joint posterior at the inspection times concentrates at the target frequencies and satisfies a Bernstein-von Mises limit with distinct interior and boundary regimes, with Gaussian and Gamma limits on different scales. The results allow arbitrary dependence across inspection times. A key technical ingredient is a fixed-interval tail bound for the coalescent-based dual death process.
ER -