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Using a Markov Chain to Construct a Tractable Approximation of an Intractable Probability Distribution
Author(s) -
HOBERT JAMES P.,
JONES GALIN L.,
ROBERT CHRISTIAN P.
Publication year - 2006
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2006.00467.x
Subject(s) - markov chain , mathematics , markov chain mixing time , construct (python library) , markov process , probability distribution , distribution (mathematics) , variable order markov model , invariant (physics) , markov model , discrete mathematics , statistics , computer science , mathematical analysis , mathematical physics , programming language
. 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.