## On the Dickey-Savage ratio estimator

**W**hile working on the estimation of our representation of the Dickey-Savage-like approximation to the Bayes factor,

with Jean-Michel Marin, we came upon a missing term in the Monte Carlo approximation

where the ‘s are simulated in one step of a two-stage or three-stage Gibbs sampler and is the full (completed) posterior derived from the pseudo-posterior . Indeed, simulating from

means that the normalising constant of this joint distribution is not but an alternative . Therefore the proper approximation to the Bayes factor should be

based on the same simulations. This obviously creates a local problem since both and are not available. However, this ratio can be estimated by regular bridge sampling from the same simulation experiment since

or from a sample from since

the later having the advantage of leading to an unbiased estimator of . which is a rather uncommon feature! When applying both evaluations to the probit example, we ended up with both estimates of the correctng factor close to 1/1.17, and very stable, which means the picture in the previous post is next to being correct! Still, there was a clear mistake in the original representation I should not have made…

October 9, 2009 at 12:58 am

[…] the Savage-Dickey paradox Following several posts on this topic, we eventually managed to write down a short note with Jean-Michel Marin, which is […]