• Bernoulli
  • Volume 13, Number 3 (2007), 641-652.

When is Eaton’s Markov chain irreducible?

James P. Hobert, Aixin Tan, and Ruitao Liu

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Consider a parametric statistical model P(dx|θ) and an improper prior distribution ν(dθ) that together yield a (proper) formal posterior distribution Q(dθ|x). The prior is called strongly admissible if the generalized Bayes estimator of every bounded function of θ is admissible under squared error loss. Eaton [Ann. Statist. 20 (1992) 1147–1179] has shown that a sufficient condition for strong admissibility of ν is the local recurrence of the Markov chain whose transition function is R(θ, dη)= Q(dη|x)P(dx|θ). Applications of this result and its extensions are often greatly simplified when the Markov chain associated with R is irreducible. However, establishing irreducibility can be difficult. In this paper, we provide a characterization of irreducibility for general state space 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 ν. Application of the main result is illustrated using two examples.

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Bernoulli, Volume 13, Number 3 (2007), 641-652.

First available in Project Euclid: 7 August 2007

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improper prior distribution local recurrence reversible Markov chain strong admissibility


Hobert, James P.; Tan, Aixin; Liu, Ruitao. When is Eaton’s Markov chain irreducible?. Bernoulli 13 (2007), no. 3, 641--652. doi:10.3150/07-BEJ6191.

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  • Billingsley, P. (1995). Probability and Measure, 2nd ed. New York: Wiley.
  • Eaton, M.L. (1982). A method for evaluating improper prior distributions. In Statistical Decision Theory and Related Topics III (S.S. Gupta and J.O. Berger, eds.) 1. New York: Academic.
  • Eaton, M.L. (1992). A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains. Ann. Statist. 20 1147–1179.
  • Eaton, M.L. (1997). Admissibility in quadratically regular problems and recurrence of symmetric Markov chains: Why the connection? J. Statist. Plann. Inference 64 231–247.
  • Eaton, M.L. (2001). Markov chain conditions for admissibility in estimation problems with quadratic loss. In State of the Art in Probability and Statistics – A Festschrift for Willem R. van Zwet (M. de Gunst, C. Klaassen and A. van der Vaart, eds.). The IMS Lecture Notes Series 36. Beachwood, Ohio: IMS.
  • Eaton, M.L. (2004). Evaluating improper priors and the recurrence of symmetric Markov chains: An overview. In A Festschrift to Honor Herman Rubin (A. Dasgupta, ed.). The IMS Lecture Notes Series 45. Beachwood, Ohio: IMS.
  • Eaton, M.L., Hobert, J.P. and Jones, G.L. (2007). On perturbations of strongly admissible prior distributions. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • Hobert, J.P., Marchev, D. and Schweinsberg, J. (2004). Stability of the tail Markov chain and the evaluation of improper priors for an exponential rate parameter. Bernoulli 10 549–564.
  • Hobert, J.P. and Robert, C.P. (1999). Eaton's Markov chain, its conjugate partner and $\mathcal{P}$-admissibility. Ann. Statist. 27 361–373.
  • Hobert, J.P. and Schweinsberg, J. (2002). Conditions for recurrence and transience of a Markov chain on $\mathbb{Z}^+$ and estimation of a geometric success probability. Ann. Statist. 30 1214–1223.
  • Meyn, S.P. and Tweedie, R.L. (1993). Markov Chains and Stochastic Stability. London: Springer.