Ok, dude, I now see your points. Actually, following your comments and a long Skype discussion with Jean-Michel from Helsingin, I modified rather seriously the description of my points about the D-S ratio representation, since they were obviously confusing… You have completely summarised my feelings about the D-S representation! Something absurd from a mathematical perspective, unless one imposes constraints that are unrelated with measure-theory. I still fail to see the connection with Wald tests, because again this is a Bayes factor and you cannot look at it as a likelihood ratio. Maybe at best as a profile likelihood ratio?

]]>I agree that what is even more puzzling in the D-S ratio is that it is of course equal to something that does only depend on the prior measure (the Bayes factor) but rewritten in a form which involve a ratio of densities! This is what I understood as your original concern.

Regarding my last comment, I did not expect the author of ‘The Bayesian choice’ to justify Bayes factors other than by the mere fact that they are Bayes factors! But my question was more to know whether this D-S ratio could be related to usual frequentist tests of equality when the sample size is large (in saying that it could perhaps be related to Wald test, I did miss however the fact that there was a nuisance parameter even under the null, so the situation is probably a bit more complicated here).

]]>Sorry, dude!, but I somehow disagree with your association of the MAP estimator with the Dickey-Savage ratio: the former is dependent on the dominating measure associated with the prior density and not so much on the version of this density (since removing a set of measure zero almost surely does not change the MAP estimate), while the Dickey-Savage ratio paradox is strictly related with the non-unicity of the density over a set of measure zero. Changing the dominating measure keeps the ratio the same.

Furthermore, I do not understand your second point. The Dickey-Savage ratio is the Bayes factor, hence it does not have any specific inferential property. It simply proposes a particular representation of the Bayes factor in terms of the prior/posterior under the alternative hypothesis. The Bayes factor being consistent, so is the Dickey-Savage ratio representation… As you rightly point out, the problem vanishes in discrete settings. To make the Dickey-Savage ratio work in continuous settings, I think imposing to the prior and posterior densities both to be continuous should be sufficient if the parameter space is connected.

In a very few days, I hope this will become clearer with the note we are completing on this (the R programme for the example is written and working). The implementation of the modified Gibbs sampler is straightforward and the numerical results coincide with other approximations to the Bayes factor, like Chib’s and the harmonic ones.

]]>This being said, the MAP estimator has a reassuring behavior when the sample size is large in that it is equivalent to the maximum likelihood estimator and hence consistent, etc. Do similar results exists for the D-S ratio? Unsurprisingly, I would expect this to be equivalent to the Wald test but has this been shown formally?

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