Just a quick note to advertise the once-revised draft of my recursive marginal likelihood estimators paper, now on astro/stat-ph: http://arxiv.org/pdf/1301.6450.pdf . The new version includes a greater emphasis on the applicability of these ideas (characterised by biased sampling) to the problem of model selection under stochastic process priors, and its effectiveness for prior-sensitivity analysis via importance sampling reweighting from the normalized bridging sequence with the appropriate Radon-Nikodym derivative. We also give some results on nested sampling (including a mathematical justification for importance sample reweighting style prior-sensitivity analysis of nested sampling output) and demonstrate both the power posteriors and partial data posteriors bridging sequences (with Gibbs sampling).
One thing this recent revision has got me thinking about is the Savage-Dickey density ratio. Where the usual SDDR is written as a ratio of densities I see a Radon-Nikodym derivative and therefore imagine a version of the SDDR for model selection in general metric spaces (possibly without Lebesgue densities). At face value I don’t see why the derivation from Marin & Robert’s (2010) paper where they specify particular versions of the conditional probability couldn’t be tweaked to become conditional probabilities on the product space with specific versions of the Radon-Nikodym derivative?
Something like this … ?