Is the Dickey-Savage ratio any valid?!
As mentioned in an earlier post on the Bayes factor, I have conceptual difficulties with the Dickey-Savage ratio… While the method is well-described in Chen, Shao and Ibrahim (20012000, pages 164-165), let me recall here that the Dickey-Savage ratio provides a representation of the Bayes factor for testing an embedded model, , with a nuisance parameter , under the assumption that the conditional prior density of under the alternative when , , is equal to the prior under the null hypothesis . In this case, we have
the prior and posterior marginal densities of under the alternative.
What bothers me with this equality (whose functional proof is quite straightforward) is that it relies on a particular version of the conditional density, i.e. that the assumption above is meaningless from a measure theoretic perspective. Given the formal definition of conditional measures and densities, they are only known defined up to a set of measure zero and the value of when is thus arbitrary (since is a fixed value that is [or should be] set before observation). Furthermore, the choice of the a version of does not impact on the choice of the version of , which also is arbitrary, so there is no cancellation of arbitrary constants in the Dickey-Savage ratio representation. This Dickey-Savage representation is therefore dependent both in its assumption and in its expression on a specific version of the conditional density . Furthermore, when is replaced with a Rao-Blackwell estimate,
this estimate also depends on the choice of a collection of versions of the conditional densities... The answer to the title of this post is therefore that, no, the Dickey-Savage representation is not valid: it simply is meaningless from a mathematical viewpoint and thus this has nothing to do with simulation issues, hence the removal of the previous sentence!
After a few more hours of thinking about this issue (in the plane to Finland), I came to realise I have a way to write down a generic valid Savage-Dickey Dickey-Savage-like ratio representation that only involves a pseudo-prior instead of imposing a meaningless constraint on the prior. Furthermore, this Dickey-Savage-like ratio representation can produce an approximation to the Bayes factor based on a corresponding single (new) sequence of simulations, without independently from the Verdinelli-Wasserman extension. Indeed, all that is needed is a Monte Carlo or MCMC sampler on with the new (pseudo-posterior) target
which is usually feasible by a completion and a Gibbs sampling algorithm. (We are currently implementing the idea with Jean-Michel Marin and should have a preprint ready pretty soon with a probit example.) The approximation based on the Dickey-Savage-like ratio representation for the Bayes factor is then
where the ‘s are simulated in one step of the a two-stage or three-stage Gibbs sampler and is the full (completed) posterior derived from the pseudo-posterior .
My conclusion is therefore that the Savage-Dickey ratio representation is a universally valid approximation technique rather than a correct mathematical representation valid under some restriction on the priors. All is well that ends well! My conclusions are therefore that (a) the Dickey-Savage ratio representation does not make sense mathematically and (b) there exists a universally valid approximation technique to the Bayes factor that relies on simulating from a well-defined pseudo-posterior and a corresponding Dickey-Savage representation, and on using an appropriate Rao-Bloackwellised estimate of a conditinal density.