Using a Markov chain to construct a tractable approximation of an intractable probability distribution

James P. Hobert, Galin L. Jones, Christian P. Robert

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

Let π denote an intractable probability distribution that we would like to explore. Suppose that we have a positive recurrent, irreducible Markov chain that satisfies a minorization condition and has π as its invariant measure. We provide a method of using simulations from the Markov chain to construct a statistical estimate of π from which it is straightforward to sample. We show that this estimate is 'strongly consistent' in the sense that the total variation distance between the estimate and π converges to 0 almost surely as the number of simulations grows. Moreover, we use some recently developed asymptotic results to provide guidance as to how much simulation is necessary. Draws from the estimate can be used to approximate features of π or as intelligent starting values for the original Markov chain. We illustrate our methods with two examples.

Original languageEnglish (US)
Pages (from-to)37-51
Number of pages15
JournalScandinavian Journal of Statistics
Volume33
Issue number1
DOIs
StatePublished - Mar 2006

Keywords

  • Burn-in
  • Gibbs sampler
  • Minorization condition
  • Mixture representation
  • Monte Carlo
  • Regeneration
  • Split chain

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