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This may involve approximation of the likelihood as well as other alterations to the algorithm itself. When we use Monte Carlo methods to fit complex models to large data sets we may need to make additional approximations. However it seems in practice a fairly stringent check Passing the test doesn’t guarantee the samples have the correct distribution, just that we cannot detect any departure. This is not simply a test for MCMC stationarityīut for the MCMC target distribution itself. By correct here we mean the Monte Carlo sampler is generating samples distributed according to the intended posterior distribution, so in the case of MCMC this means the algorithm is both correctly implemented and the "burn-in" is large enough to give representative samples from the target. The test is usually made in the non-parametric form suggested by Cook in 2006.
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In a paper from 2004 with the snappy title "Getting it Right", Geweke uses this property to test an MCMC sampler targeting a given posterior distribution. In this ideal setting we have exact coverage. Title: Calibrating approximate Bayes coverageĪbstract: Consider Bayesian inference and suppose the prior and likelihood are both exactly correct - that is, nature really does draw the parameter theta according to pi(theta) and the data y according to p(y|theta). If we make a Bayesian credible set C_y(alpha) of size 1-alpha from the posterior in the usual way then it covers the true value of the parameter theta with probability 1-alpha.