# When is Eaton's Markov chain irreducible - Mathematics > Statistics Theory

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Abstract: Consider a parametric statistical model $P\mathrm{d}x|\theta$ and animproper prior distribution $ u\mathrm{d}\theta$ that together yield aproper formal posterior distribution $Q\mathrm{d}\theta|x$. The prior iscalled strongly admissible if the generalized Bayes estimator of every boundedfunction of $\theta$ is admissible under squared error loss. Eaton Ann.Statist. 20 1992 1147-1179 has shown that a sufficient condition for strongadmissibility of $ u$ is the local recurrence of the Markov chain whosetransition function is $R\theta,\mathrm{d}\eta=\intQ\mathrm{d}\eta|xP\mathrm {d}x|\theta$. Applications of this result and itsextensions are often greatly simplified when the Markov chain associated with$R$ is irreducible. However, establishing irreducibility can be difficult. Inthis paper, we provide a characterization of irreducibility for general statespace Markov chains and use this characterization to develop an easily checked,necessary and sufficient condition for irreducibility of Eaton-s Markov chain.All that is required to check this condition is a simple examination of $P$ and$ u$. Application of the main result is illustrated using two examples.

Author: ** James P. Hobert, Aixin Tan, Ruitao Liu**

Source: https://arxiv.org/