Bayes Factors via Serial Tempering

Let M be a finite or countable set of models (here we only deal with finite M but Bayes factors make sense for countable M). For each model m ∈ M we have the prior probability of the model pri(m). It does not matter if this prior on models is unnormalized. Each model m has a parameter space Θm and a prior g(θ | m), θ ∈ Θm The spaces Θm can and usually do have different dimensions. That’s the point. These within model priors must be normalized proper priors. The calculations to follow make no sense if these priors are unnormalized or improper. Each model m has a data distribution