Markov chain Monte Carlo

We have seen how to sample out of distributions that have a convenient transform or inverse CDF. We have also seen how to simulate stories of distributions and to use clever tricks like thinning. We now turn to the question, how can we sample out of an arbitrary distribution for which I an write the probability density function? This has immediate applications in Bayesian inference, since it will enable us to sample out of a posterior distribution \(g(\theta\mid y)\) in order to make sense of it.

Dominant among techniques to perform sampling out of an arbitrary distribution is Markov chain Monte Carlo.