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MC's for MCMC'ists
Author(s) -
Nummelin Esa
Publication year - 2002
Publication title -
international statistical review
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.051
H-Index - 54
eISSN - 1751-5823
pISSN - 0306-7734
DOI - 10.1111/j.1751-5823.2002.tb00361.x
Subject(s) - markov chain monte carlo , markov chain , mathematics , central limit theorem , estimator , markov chain mixing time , empirical distribution function , variable order markov model , random variable , markov property , bayesian probability , markov model , statistics
Summary We develop a minimum amount of theory of Markov chains at as low a level of abstraction as possible in order to prove two fundamental probability laws for standard Markov chain Monte Carlo algorithms: 1. The law of large numbers explains why the algorithm works: it states that the empirical means calculated from the samples converge towards their “true” expected values, viz. expectations with respect to the invariant distribution of the associated Markov chain (=the target distribution of the simulation). 2. The central limit theorem expresses the deviations of the empirical means from their expected values in terms of asymptotically normally distributed random variables. We also present a formula and an estimator for the associated variance.