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Markov chain Monte Carlo (MCMC) algorithms are an indispensable tool for performing Bayesian inference. This review discusses widely used sampling algorithms and illustrates their implementation on a probit regression model for lupus data. The examples considered highlight the importance of tuning the simulation parameters and underscore the important contributions of modern developments such as adaptive MCMC. We then use the theory underlying MCMC to explain the validity of the algorithms considered and to assess the variance of the resulting Monte Carlo estimators.
Craiu et al. (Mon,) studied this question.