An Introduction to Bayesian MCMC Algorithms

Ziheng Yang and Mario dos Reis

Yang, Z. 2016. Bayesian phylogenetic inference. Pages 137-140 in Encyclopedia of Evolutionary Biology (Kliman, ed.) Elsevier. 4-page pdf
R code for a 1-D MCMC algorithm: Bayesian estimation of the distance (d) between the human-orangutan mitochondrial 12S rRNA genes under the Jukes-Cantor mode. This is the example mentioned in the paper. This is also Example 7.1 in Yang (2014 Molecular Evolution: A Statistical Approach, p.216).
The R code has four parts.
Part A calculates the posterior of the distance d using numerical integration, illustrating the computational challenges of Bayesian inference.
Part B implements the MCMC algorithm, with the proposal being a sliding window around the current state with reflection. The exercise illustrates the impact of the window size on the acceptance proportion and on efficiency.
Part C illustrates such concepts as autocorrelation, efficiency, effective sample size (ESS), and thinning.
Part D illustrates a few techniques for diagonising MCMC, such as trace plots and comparison between multiple runs.
R code for a 2-D MCMC algorithm: Bayesian estimation of rate (r) and time (t) using the human-orang mitochondrial 12S rRNA genes. Part A illustrates Bayesian computation using numerical integration. Part B is an MCMC algorithm. Each MCMC iteration involves two 1-D sliding windows moves, one that changes r and another that changes t.