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[log in to unmask]">Andrew Gelman's post about inverse-gamma prior for variance parameters (gamma distribution for precision) had me puzzled for some time. I was quite surprised to see how much less informative sigma.theta~dunif(0,1000) was compared to the more common tau.theta~dgamma(.0001,.0001). This uninformativeness expresses itself not only in a diffuse (high variance) posterior distribution of sigma, but also in a theta values as well. I admit that I was beginning to dispair for the future of our beloved gamma priors. It was no real comfort that the uniform sigma was a kluge, that required setting the range of the uniform to match the data, since if the true standard deviation was in the thousands, we could just bump it to dunif(0,1.0E6) or wider. We could in fact set this upper limit to the maximum floating point value for the system to provide a less artificial prior. The problem with this uniform prior on sigma is that you can produce even more uninformative priors in this same method. If a uniform prior on sigma is a good prior, then why not a uniform prior on sigma^2? It turns out that a uniform variance prior produces a sigma.theta distribution that has an even wider (higher variance) posterior, and higher variance on the theta posteriors as well. If we take the variance of sigma.theta, mu.theta, and theta posteriors to be a good metric of uninformativeness, then a uniform variance prior is certainly less informative than a uniform standard deviation prior. Now if we think that a uniform distribution on sigma^2 is a good idea, then you can imagine how good a uniform distribution on sigma^4 or sigma^6 will be. Unfortunately, the simulations either crash or are saved by bumping up against the upper limit on the uniform distribution (making the prior intrinsically informative). But we can see from a uniform distribution on sigma^2.5 that the 'uninformativeness' of the sigma prior increases with the exponent. The problem with this uniform approach is that it creates an artificially fat tail (prior probabilities increase away from zero) which artificially inflates the resulting sigma.theta posterior means and variances. This produces a corresponding increase in the variances of the resulting theta posteriors. Given its kluge nature, the fact that we can arbitrarily increase the vagueness of sigma by increasing the exponent of the uniform prior, and that there is no 'obvious' choice for the value of this exponent, we should be very cautious about using cropped uniform (improper) priors for variance parameters. In light of the above, my doubts about the vagueness of the inverse-gamma prior were the result of a comparison with a prior that was not in fact uninformative, but rather artificially promoted vague posteriors. If we rule out chopped improper (uniform) distributions, then it is hard to beat vague gammas (i.e. gammas with very small value parameters) for vagueness. - Finn Krogstad P.S. This is not to say that chopped uniform priors do not have their place. Their computational simplicity can be useful, especially when we don't have conjugacy. But in general, the realization that we can always make a more vague uniform prior should make us uncomfortable with the vagueness of chopped uniform priors.
Hi All! Andrew Gelman's message gives me a nice excuse to share some concerns that I have always had with you all. I think that he problem might not be caused only by the use of the inverse-gamma . It has repeteadly been reported that using a "vague proper prior" which is "close" to being "non-informative" in the sense of being "close" to an improper, non-informative prior is a HORRIBLE idea when this improper prior would produce an IMPROPER posterior. This is the case with the variance components problem: the prior 1/sigma2 for the variance component (corresponding to an inverse-gamma(0,0) ) yields and improper posterior, and any attempt to get "vague" with an inverse-gamma with small values of the hyper-parameters is not a good idea, and will not work. On the other hand, a constant prior on sigma yields a proper posterior; hence, approximating this improper constant prior with a "chopped" proper uniform should work much better. This, I think, should be a subject of some concern and lots of work, since the use of the dangerous kind of "vague proper priors" is quite frequent. Susie Bayarri -- ******************************** M.J. bayarri Univ. Valencia, Dept. of Statistics and O.R. Av. Dr. Moliner 50 46100 Burjassot, Valencia, Spain Ph: (34) 96 354 4309 (34) 96 354 4709 ********************************
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