Estimation in hidden Markov models via efficient importance sampling

Given a sequence of observations from a discrete-time, finite-state hidden Markov model, we would like to estimate the sampling distribution of a statistic. The bootstrap method is employed to approximate the confidence regions of a multi-dimensional parameter. We propose an importance sampling formula for efficient simulation in this context. Our approach consists of constructing a locally asymptotically normal (LAN) family of probability distributions around the default resampling rule and then minimizing the asymptotic variance within the LAN family. The solution of this minimization problem characterizes the asymptotically optimal resampling scheme, which is given by a tilting formula. The implementation of the tilting formula is facilitated by solving a Poisson equation. A few numerical examples are given to demonstrate the efficiency of the proposed importance sampling scheme.

💡 Research Summary

The paper addresses the problem of estimating the sampling distribution of a statistic derived from observations generated by a discrete‑time, finite‑state hidden Markov model (HMM). In many applied settings—such as speech recognition, bioinformatics, or econometrics—researchers need confidence regions for multi‑dimensional parameters, but the dependence structure of HMMs makes standard bootstrap procedures inefficient. The authors propose an importance‑sampling scheme that dramatically improves the efficiency of bootstrap‑based inference for HMMs.

The core idea is to view the conventional bootstrap resampling rule (which simply draws new state‑observation sequences using the estimated model parameters (\hat\theta)) as a baseline within a locally asymptotically normal (LAN) family of probability measures. By perturbing the baseline distribution with a small parameter shift (\delta) around (\hat\theta), one obtains a family ({P_{\hat\theta+\delta}}) whose asymptotic variance for the statistic of interest can be expressed as a quadratic form (\delta^{\top} I(\hat\theta),\delta), where (I(\hat\theta)) is the Fisher information matrix of the HMM. Minimizing this quadratic form over (\delta) yields the asymptotically optimal perturbation.

Mathematically, the optimal shift is (\delta^{}=I(\hat\theta)^{-1}\nabla\ell(\hat\theta)), where (\nabla\ell(\hat\theta)) denotes the gradient of the log‑likelihood (or, more generally, the influence function of the statistic). Substituting (\delta^{}) into the likelihood ratio produces a tilting weight
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