The Metropolis algorithm, and its generalization (Metropolis-Hastings algorithm) provide elegant methods for obtaining sequences of random samples from complex probability distributions. When I first read about modern MCMC methods, I had trouble visualizing the convergence of Markov chains in higher dimensional cases. So, I thought I might put together a visualization in a two-dimensional case.
I’ll use a simple example: estimating a population mean and standard deviation. We’ll define some population level parameters, collect some data, then use the Metropolis algorithm to simulate the joint posterior of the mean and standard deviation.
Then, to visualize the evolution of the Markov chains, we can make plots of the chains in 2-parameter space, along with the posterior density at different iterations, joining these plots together using ImageMagick (in the terminal) to create an animated .gif: