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6 - Markov Chain Monte Carlo

Published online by Cambridge University Press:  27 July 2023

Daniel Sanz-Alonso
Affiliation:
University of Chicago
Andrew Stuart
Affiliation:
California Institute of Technology
Armeen Taeb
Affiliation:
University of Washington
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Summary

In this chapter we study Markov chain Monte Carlo (MCMC), a methodology that delivers approximate samples from a given target distribution π. The methodology applies to settings in which π is the posterior distribution in (1.2), but it is also widely used in numerous applications beyond Bayesian inference. As with Monte Carlo and importance sampling, MCMC may be viewed as approximating the target distribution by a sum of Dirac masses, thus allowing the approximation of expectations with respect to the target. Implementation of Monte Carlo presupposes that independent samples from the target can be obtained. Importance sampling and MCMC bypass this restrictive assumption: importance sampling by appropriately weighting independent samples from a proposal distribution, and MCMC by drawing correlated samples from a Markov kernel that has the target as invariant distribution.

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Publisher: Cambridge University Press
Print publication year: 2023

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