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Anon.
Posts:
379
Registered:
6/2/05


Re: different priors (flat, uniform, etc)
Posted:
Nov 5, 2006 7:50 AM


Herman Rubin wrote: > In article <_u%0h.40667$nQ2.12889@reader1.news.jippii.net>, > Anon. <bob.ohara@NOSPAMhelsinki.fi> wrote: >> Reef Fish wrote: >>> David Winsemius wrote: >>>> "Reef Fish" <large_nassua_grouper@yahoo.com> wrote in >>>> news:1161965644.243372.111030@i3g2000cwc.googlegroups.com: > >>>>> For Bayesian Inference on the parameter p of a Binomial distribution >>>>> or a Bernoulli Process, the beta distribution is a member of the >>>>> conjugate prior family  meaning both the prior AND posterior >>>>> belongs to the same distribution family  Beta. > >>>>> The uniform distribution on (0,1) is a Beta distribution with >>>>> parameters (1,1) and is an INFORMATIVE prior. >>>> Can we hear a bit more about how is Beta(1,1) is an informative prior for a >>>> binomial problem? > >>> It CHANGES the likelihood function to form the posterior distr. > >> But what does this mean? I guess you could mean something similar to >> the way Fisher treated likelihood: he waved his Fiducial wand, and the >> conditioning magically reversed. Of course, the Bayesian version does >> this formally. > >> The problem with this interpretation is that any prior will have the >> same effect, so there would be no such thing as a noninformative prior. >> As noninformative priors do exist, and are discussed in the >> literature, they do exist. > > Is there such a thing as a noninformative prior? I see no > justification for such, and good reasons not to use such. > I take a descriptive approach to definitions, so there is such a thing as a noninformative prior, simply because people use the term. Whether they should is a matter that could be discussed endlessly, and it certainly wasn't my aim to take a firm stand either way in this thread.
> For some problems, invariant priors are used, with the best > invariant prior being the right invariant Haar measure for > the transformation group. Priors should be looked upon as > weight functions, rather than belief, and hence can have an > infinite integral. The usual argument given for invariant > priors is that if one has a location problem, it matters not > where the origin is located, or if one has a scale problem, > the units do not matter. > > Now it is correct that the same results should be obtained > if the units are inches or meters, but this does not mean > that the inference should be the same if the numbers given > are the same. There are invariant problems in which there > are priors giving uniformly better results than invariant > priors, and these are not "unusual"; estimating the > covariance matrix of a multivariate normal is there already. > >> Noninformative priors are generally defined as priors which only add a >> small amount of information, as compared to the likelihood. How does >> the beta(1,1) shape up? > > What does this mean? If the sample size is large enough, > and the dimension is small enough, and the prior is "smooth", > it makes essentially no difference. > Indeed: but of course that isn't always the case, and I was trying to pin down a specific comment by Reef Fish.
>> For the binomial, the likelihood (up to a normalising constant) is: > >> L(p r) = p^n (1p)^(Nn) > >> The pdf of a beta distribution is: > >> P(p) = K p^(alpha1) (1p)^(beta1) > >> (where K is a normalising constant) so the posterior is > >> P(Pr) = K_p p^(n+alpha1) (1p)^(Nn+beta1) > >> For a beta(1,1), this becomes: > >> P(Pr) = K_p p^(n) (1p)^(Nn) > >> i.e. algebraically the same as the likelihood. In other words, it >> doesn't add any information to the likelihood. This is pretty much >> definitive of a "noninformative prior". > > So should one use a beta(1,1) or a beta(.5,.5) or a beta(0,0)? > This latter would use the density 1/(p  p^2), which is the > reciprocal of the information? This and its square root have > been suggested, and in the case of an invariant problem, will > automatically give an invariant procedure, which may be quite > bad throughout the parameter space. > So, the invariant approach may not be the best in all cases. I guess almost any "noninformative", "vague", "objective" approach to developing priors will break down in some circumstances.
Bob
 Bob O'Hara Department of Mathematics and Statistics P.O. Box 68 (Gustaf Hällströmin katu 2b) FIN00014 University of Helsinki Finland
Telephone: +3589191 51479 Mobile: +358 50 599 0540 Fax: +3589191 51400 WWW: http://www.RNI.Helsinki.FI/~boh/ Journal of Negative Results  EEB: www.jnreeb.org



