The model evidence is a vital quantity in the comparison of statistical
models under the Bayesian paradigm. This paper presents a review of commonly
used methods. We outline some guidelines and offer some practical advice. The
reviewed methods are compared for two examples; non-nested Gaussian linear
regression and covariate subset selection in logistic regression.

This paper presents a Markov chain Monte Carlo method to generate approximate
posterior samples in retrospective multiple changepoint problems where the
number of changes is not known in advance. The method uses conjugate models
whereby the marginal likelihood for the data between consecutive changepoints
is tractable. Inclusion of hyperpriors gives a near automatic algorithm
providing a robust alternative to popular filtering recursions approaches in
cases which may be sensitive to prior information. Three real examples are used
to demonstrate the proposed approach.

The method of tempered transitions was proposed by Neal (1996) for tackling
the difficulties arising when using Markov chain Monte Carlo to sample from
multimodal distributions. In common with methods such as simulated tempering
and Metropolis-coupled MCMC, the key idea is to utilise a series of
successively easier to sample distributions to improve movement around the
state space. Tempered transitions does this by incorporating moves through
these less modal distributions into the MCMC proposals.