Asymptotics and finite sample bounds for prediction and smoothing in Wright-Fisher hidden Markov models
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.