An analytical study of several markov chain monte carlo estimators of the marginal likelihood

An approach to calculating the marginal likelihood (ML) by the Gibbs stopper is examined. The moments of the estimator are investigated assuming a normal posterior distribution. The analytical expectation and variance for a variety of ML estimators based on the Gibbs stopper, as well as based on another approach proposed by Chib, are derived and are compared. It is found that even in relatively simple situations (e.g., bivariate and multivariate normal posteriors), an estimator can have infinite variance, especially when the parameters are highly correlated. Some fixes to this phenomena are proposed and asymptotic properties of the estimators are discussed. The estimators are applied to real data.