We prove that under certain mild assumptions, the entropy rate of a hidden Markov chain, observed when passing a finite-state stationary Markov chain through a discrete-time continuous-output channel, is jointly analytic as a function of the input Markov chain parameters and the channel parameters. In particular, as consequences of the main theorems, we obtain analyticity for the entropy rate associated with representative channels: Cauchy and Gaussian.
To the best of our knowledge, the results in this paper, together with those in @cite_0 , are among the first results establishing analyticity of continuous-state hidden Markov chains. Given the interest in the counterpart results for the discrete-state setting, we expect that such results will be of significance in the continuous-state setting as well.
@cite_0
2159877461 2070178449 1996769950 2028920027
We prove that under mild positivity assumptions, the entropy rate of a continuous-state hidden Markov chain, observed when passing a finite-state Markov chain through a discrete-time continuous-output channel, is analytic as a function of the transition probabilities of the underlying Markov chain. We further prove that the entropy rate of a continuous-state hidden Markov chain, observed when passing a mixing finite-type constrained Markov chain through a discrete-time Gaussian channel, is smooth as a function of the transition probabilities of the underlying Markov chain. We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of σ-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365―393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space. Abstract We address the issue of order identification for hidden Markov models with Poisson and Gaussian emissions. We prove information-theoretic BIC-like mixture inequalities in the spirit of Finesso [1991. Consistent estimation of the order for Markov and hidden Markov chains. Ph.D. Thesis, University of Maryland]; Liu and Narayan [1994. Order estimation and sequential universal data compression of a hidden Markov source by the method of mixtures. Canad. J. Statist. 30(4), 573–589]; Gassiat and Boucheron [2003. Optimal error exponents in hidden Markov models order estimation. IEEE Trans. Inform. Theory 49(4), 964–980]. These inequalities lead to consistent penalized estimators that need no prior bound on the order. A simulation study and an application to postural analysis in humans are provided. We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovasz and Kannan, can be refined to apply to the maximum relative deviation |p n (x,y) π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.