MCMC

Abstract

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.

Before I had succeeded in solving my problem analytically, I had endeavoured to do so empirically. The material used was a correlation table containing the height and left middle finger measurements of 3000 criminals, from a paper by W. R. Macdonell (Biometrika, Vol. I, p. 219). The measurements were written out on 3000 pieces of cardboard, which were then very thoroughly shuffled and drawn at random. As each card was drawn its numbers were written down in a book which thus contains the measurements of 3000 criminals in a random order. Finally each consecutive set of 4 was taken as a sample—750 in all—and the mean, standard deviation, and correlation of each sample determined.

William Searly Gosset (‘Student’), 1908

The original version of this chapter was revised. A correction to this chapter is available at https://doi.org/10.1007/978-3-319-54274-4_11.