We introduce in this chapter a numerical method, Markov Chain Monte Carlo (MCMC), which will allow us to find an accurate estimate of the marginal posterior distributions without the need of solving the integrals required for marginalisation. The formal justification of the MCMC procedure is complex and out of the scope of this book, but we can intuitively understand how it works. We will see with some detail the most common MCMC procedure used in animal production, Gibbs sampling, and we will sketch other common MCMC procedures. This is an active research area with continuous new developments, but it is not a part of Bayesian statistics. MCMC is only a numerical tool for approximating marginal posterior distributions without solving the integrals that stopped in the past the practical development of Bayesian statistics in many fields of knowledge.
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