Hi all. As always, I would like to thank the Bugs crew for creating
this wonderful piece of free software!
I have only a slight comment on David's book extract: it says that
Gelman et al. recommend a uniform prior
distribution on tau^2. Actually, we use a uniform prior distribution on
tau. I think there has been some confusion
on this dating back to the Rubin (1981) paper. There's more on this
prior distribution issue in Appendix C
of the 2nd edition of our book.
A related issue is that, when you use the noninformative uniform prior
distribution on tau, it tends to "bias" things
slightly towared large values of tau (i.e., less shrinkage). In fact,
if the number of "random effects" is 1 or 2,
the posterior distribution for tau still has infinite mass near
infinity, and so the hierarchical analysis does no
shrinkage at all. For relatively small numbers of "random effects",
e.g., 3 or 4, only a small amount of shrinkage
is typically done; i.e., the hierarchical analysis tends to be
"conservative" in the sense of producing inferences
that are close to what would be obtained from a no-pooling model.
Andrew
Kenneth Rice wrote:
>The file concerning gamma priors (180K) is at
>
>http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/tauprior.pdf
>
>and not the site originally given.
>
>Ken Rice
>Bugs list manager
>
>------------- Begin Forwarded Message -------------
>
>Dear All
>
>Sorry to be so late in contributing to this vital discussion.
>
>a) I agree with the discussants who point out that priors like
>Gamma(0.001, 0.001) can be dangerous when used for random effects
>precisions, and cannot really be claimed to be 'non-informative'
>
>b) A number of options are available. I take the liberty of plugging a
>forthcoming book (
>http://www.wileyeurope.com/WileyCDA/WileyTitle/productCd-0471499757.html )
> which contains a discussion of these issues. A draft of the relevant
>pages can be downloaded from
>http://mrc-bsu.cam.ac.uk/bugs/winbugs/tauprior.pdf
>
>c) As others have said, if you want to make any claim as to
>'non-informativeness', then you should be placing a prior directly on an
>interpretable parameter. The random effects precision certainly is not
>such a parameter, but the random effects standard deviation (which I shall
>call tau) is.
>
>d) The plots in the extract show that the prior on tau implied by
>Gamma(0.001, 0.001) on 1/tau^2 is not, in fact, so ridiculous for random
>effects logistic and Poisson models (although it may be for Normal). I
>think this is the (perhaps fortuitous) reason why it has not caused too
>many problems in the past.
>
>e) My personal preference is now to use a uniform prior on tau, or a
>half-normal on tau when I want to emphasise smaller values.
>
>f) We must acknowledge that the Gamma(0.001, 0.001) prior features in many
>BUGS examples and so has been taken up by many users. It was originally
>used because, back in the early days of BUGS, conjugacy was a great
>help. This is no longer the case and, to be completely honest, I think we
>should have redone all the examples that use Gamma(0.001, 0.001) priors
>before the release of WinBUGS 1.4 to check the sensitivity and see whether
>a uniform on tau is OK. But I'm afraid we didn't. We certainly would like
>to avoid any suggestion that this is the recommended prior.
>
>g) Is there some generous person out there who would be willing to conduct
>a class exercise to rerun all those examples with a variety of priors? The
>code could be adapted from the 'prior sensitivity trick' in the 1.4
>manual. If you posted up the results, you and your class could get
>whatever meagre credit might be due.
>
>david spiegelhalter
>
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