In response to Finn Krogstad's and Andrew Millard's postings:
While Finn's point is well taken, it is nonetheless true that some scales
are more meaningful than others. Thus, a flat or other prior for
variation on one given scale may be a particularly meaningful
specification of indifference about the degree of variation. I personally
find standard deviation the most meaningful scale because it alone is on
the same scale as the original measurements. However, there are pretty
good arguments for other scales, notably precision, which a scale that is,
linearly increasing in the number of observations in simple normal
problems.
Another concern I have is that the gamma or inverse gamma(epsilon,
epsilon) prior can produce undesirable posterior artifacts, like
bimodality. For example, in the balanced one-way random effects model,
the restricted likelihood is always unimodal in the two unknown variances,
while the corresponding marginal posterior is easily made bimodal using
inverse gamma priors for the two variances. Given that the restricted
likelihood is identical to the marginal posterior for a flat prior on both
variances and a flat prior on the overall mean, it follows that the prior
is the sole cause of this marginal posterior bimodality. Bimodality of
the posterior for this problem is explored in a recent JRSSB paper:
Liu J, Hodges JS. Posterior bimodality in the balanced one-way random
effects model. J. Royal Stat. Soc., Ser. B, 65:247-255, 2003
I do not mean to advocate flat priors for the variances in this problem,
only to emphasize that we have a lot more work to do before we can claim
to understand the consequences of priors for these models.
Best wishes to you all,
Jim Hodges
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