Hidden Markov model for parameter estimation of a random walk in a Markov environment

Abstract

We focus on the parametric estimation of the distribution of a Markov environment from the observation of a single trajectory of a onedimensional nearest-neighbor path evolving in this random environment. In the ballistic case, as the length of the path increases, we prove consistency, asymptotic normality and efficiency of the maximum likelihood estimator. Our contribution is two-fold: we cast the problem into the one of parameter estimation in a hidden Markov model (HMM) and establish that the bivariate Markov chain underlying this HMM is positive Harris recurrent. We provide different examples of setups in which our results apply, in particular that of DNA unzipping model, and we give a simple synthetic experiment to illustrate those results.

Fundamental limits for learning hidden Markov model parameters

“…Finally, let us mention that departure from the independence assumption has been noted to allow much better possible learning also in HMM settings free from the assumption that the Markov chain has a finite state space [17,6] (at the expense of stricter assumptions on the emission distributions), and also in other problems including dynamic networks [30,7], image denoising [31], and deconvolution [16].…”

The heavy range of randomly biased walks on trees

“…Recently more attention has been paid to the one dimensional case: the aim was to provide a parametric estimation of the distribution of the environment with the help of a single trajectory of random walk (X n ) n∈N ([CFL + 14], [FLM14], [FGL14], [CFLL16]). When the environment is a Markov chain a parametric approach can also be found in [ALM15]. The problem of non-parametric estimation has been studied in [DL17], the aim of the result we present below is to extend their studies to the more delicate case of randomly biased random walks on supercritical Galton-Watson trees.…”

Some recent advances in random walks and random environments

“…More recently, [9,10,15,16] considered the random walk on Z and investigated the problem in a parametric framework. The case of Markovian environment has also been investigated in [3]. Although very interesting, this approach suffers several drawbacks.…”