Marginal Likelihood Calculation with MCMC Methods

Abstract

Markov Chain Monte Carlo (MCMC) methods have revolutionised Bayesian data analysis over the years by making the direct computation of posterior probability densities feasible on modern workstations. However, the calculation of the prior predictive, the marginal likelihood, has proved to be notoriously difficult with standard techniques. In this chapter a method is presented that lets one calculate the marginal likelihood using nothing but the results from standard MCMC algorithms, like Metropolis-Hastings. This new method is compared to other methods like nested sampling, and outperforms the latter in some cases. One of the toy problems considered in this chapter is the analysis of mock pulsar timing data, as encountered in pulsar timing array projects. This method is expected to be useful as well in other problems in astrophysics, cosmology and particle physics.