# Sampler method to choose in Monte Carlo Markov chain estimation

When estimating the posterior via MCMC, are there guidelines on the best sampling method to use depending on the nature of the model?

There are a variety of forms of MCMC - the Gibbs sampler, the Metropolis algorithm, and the Metropolis-Hastings algorithm for example. Gilks et al. (1996) has a nice discussion of the MCMC variations.

My particular use-case involves estimating a posterior by mixing observed sample security returns with a prior equilibirum model of security returns. Return series are often heteroskedastic and have cross-sectional correlations and are non-stationary across market regimes.

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At a secondary level, MCMC algorithms can be compared in terms of speed of convergence: if I start from an arbitrary $x_0$, for which $t$ is $x_t$ exactly distributed from the target density $f$? approximately distributed from the target density $f$? This is a much harder question and very little can be said for a real complex model that does require the use of MCMC... There exist deep and involved theoretical results that were obtained by researchers like Richard Tweedie, Gareth Roberts, Jeff Rosenthal, Éric Moulines, Jim Hobert, and many others (if you want to look at the literature), but it is extremely complicated to turn those results into a comparison criterion when several MCMC algorithms are competing.