Markov Chain Monte Carlo Sampling (MCMC)

The Bayesian approach to parameter estimation requires modellers to make a major mental shift: we no longer aim to find a single ‘best’ parameter vector—instead we aim to determine the posterior probability distribution for the parameters. Only the full probability distribution adequately represents our state of knowledge. Although this shift in thinking has made rigorous uncertainty quantification possible, it has also created computational problems. When models have many parameters, the distribution will be high-dimensional and may not be available in closed form, i.e. we cannot derive a formula for it. Often, all we can do is generate a sample from the distribution, and even that can be difficult. A solution for the problem of sampling from probability distributions that are not in closed form was provided by Metropolis et al. (J Chem Phys 21:1087–1092, 1953). They introduced the so-called Markov Chain Monte Carlo (MCMC) method. MCMC is the workhorse of computational Bayesian statistics, and by now, many different MCMC algorithms exist.