Last week I posted a question about the Surgical Institutional Ranking
example in BUGS. My original question is below. In summary, I was
asking whether a standard improper prior on the parameters would lead
to an improper posterior, and whether the diffuse prior that was used
in the example would lead to poor behaviour in the posterior because
of this.
I received helpful responses from several people.
Susie Bayarri has done some work on this and presented it at the ISBA
meeting in Crete.
Martyn Plummer pointed out that in the information about the
dispersion parameter tau in the over-dispersed binomial model comes
from the over-dispersion in the observed responses, and suspected that
there would be little prior sensitivity. He pointed out his paper on
the topic (available at
<http://calvin.iarc.fr/~martyn/papers/sensitivity.ps>), and the paper
by Natajaran and Kass (2000, JASA) on the issue.
Paul Hewson also pointed out the Natajaran and Kass (2000) paper.
The Natajaran and Kass paper does make clear that the standard 1/sigma
prior for the dispersion of the p's will give an improper posterior.
They suggested a shrinkage prior to use instead.
So my tentative conclusion is that even though the improper prior
*would* lead to an improper posterior, for this data that doesn't
indicate a problem with the prior used in BUGS. I have asked a
student to investigate exactly what the consequences of the improper
posterior would be. If we find anything more, I'll report it to the
list.
Thanks to everyone for the information they provided.
Duncan Murdoch
>In the WinBUGS example "Surgical Institutional Ranking", the random effects model looks like this:
>
>model
> {
> for( i in 1 : N ) {
> b[i] ~ dnorm(mu,tau)
> r[i] ~ dbin(p[i],n[i])
> logit(p[i]) <- b[i]
> }
> pop.mean <- exp(mu) / (1 + exp(mu))
> mu ~ dnorm(0.0,1.0E-6)
> sigma <- 1 / sqrt(tau)
> tau ~ dgamma(0.001,0.001)
> }
>
>This is a hierarchical model with binary responses, a normal prior on the logits of the response rates for each, and a diffuse (but proper) hyperprior on the parameters of the normal.
>
>If we actually used an improper prior on the normal precision, would this lead to an improper posterior? I've read that it does with a normal mixture model, so I'd guess so, but I'm not sure.
>
>Assuming it does, shouldn't that make the results here quite sensitive to the actual choice of proper prior? E.g. I'd expect that if I'd used dgamma(0.0001,0.0001) I'd see quite different results.
>
>All of the above reasoning makes sense to me, and indeed, when I make that change to the prior for tau, I see substantially different results for the mean and s.d. of the posterior for tau (though not much as much difference in the quantiles).
>
>HOWEVER, all of the other parameters (including sigma) come up with results that are very close to the ones with the original prior. This leads me to wonder whether I should worry about that fact that the posterior is nearly improper or not.
>
>Could someone with more experience in this comment, please?
>
>Duncan Murdoch
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