• Bernoulli
  • Volume 25, Number 2 (2019), 877-901.

Bayesian consistency for a nonparametric stationary Markov model

Minwoo Chae and Stephen G. Walker

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We consider posterior consistency for a Markov model with a novel class of nonparametric prior. In this model, the transition density is parameterized via a mixing distribution function. Therefore, the Wasserstein distance between mixing measures can be used to construct neighborhoods of a transition density. The Wasserstein distance is sufficiently strong, for example, if the mixing distributions are compactly supported, it dominates the sup-$L_{1}$ metric. We provide sufficient conditions for posterior consistency with respect to the Wasserstein metric provided that the true transition density is also parametrized via a mixing distribution. In general, when it is not be parameterized by a mixing distribution, we show the posterior distribution is consistent with respect to the average $L_{1}$ metric. Also, we provide a prior whose support is sufficiently large to contain most smooth transition densities.

Article information

Bernoulli, Volume 25, Number 2 (2019), 877-901.

Received: April 2016
Revised: September 2017
First available in Project Euclid: 6 March 2019

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Zentralblatt MATH identifier

Kullback–Leibler support mixtures nonparametric Markov model posterior consistency Wasserstein metric


Chae, Minwoo; Walker, Stephen G. Bayesian consistency for a nonparametric stationary Markov model. Bernoulli 25 (2019), no. 2, 877--901. doi:10.3150/17-BEJ1007.

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