We study prediction and smoothing in hidden Markov models with a latent signal given by a multi-type Wright-Fisher diffusion and discrete-time categorical observations, motivated by repeated-sampling time-series settings, including temporally binned ancient-DNA data, in which noisy frequency counts are recorded at finitely many times. Our focus is on the exact Bayesian predictive and smoothing distributions available under parent-independent mutation, in relation to their large-sample targets under repeated within-time sampling. For a fixed collection-time grid and diverging within-time sample sizes, we show that the exact Wright-Fisher predictive and smoothing distributions converge in total variation to the corresponding population transition and bridge laws at the limiting neighboring frequencies. We then derive explicit finite-sample control for the predictive law and a corresponding finite-sample bound for the marginal smoother. Finally, at the inspection times, we show that the joint conditional law concentrates at the target frequencies and that its active coordinates are asymptotically Gaussian, while coordinates with zero true frequencies converge to Gamma limits at faster rates. Our regime imposes no restriction on dependence across inspection times beyond within-time sampling. The analysis rests on a fixed-interval tail bound for Kingman's coalescent block-counting process, which is of independent interest.
Luigi M. Malgieri, F. Ascolani, M. Giordano et al.· 0 citations
We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a standard Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024). It combines refined Laplace-type estimates for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincar\'e inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation with a logistic parametrisation, Poisson intensity estimation with an exponential link, and the Gaussian white-noise model, yielding minimax contraction rates in Sobolev norms across all three settings. In particular, these yield optimal recovery of density score functions and derivatives of Poisson intensities.
Emanuele Dolera, Stefano Favaro, M. Giordano· 0 citations
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