Talk:Prior probability

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The statement: "For example, Edwin T. Jaynes has published an argument [a reference here would be useful] based on Lie groups that if one is so uncertain about the value of the aforementioned proportion p that one knows only that at least one voter will vote for Kerry and at least one will not, then the conditional probability distribution of p given one's state of ignorance is the uniform distribution on the interval [0, 1]." seems highly improbable, unless Jaynes posthumously thought that the elctorate of the United States was infinite. --Henrygb 21:51, 19 Feb 2005 (UTC)

I think I know what's being referred to here. Jaynes wrote a paper, "Prior Probabilities," [IEEE Transactions of Systems Science and Cybernetics, SSC-4, Sept. 1968, 227-241], which I have reprinted in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, Dordrecht, Holland: Reidel Publishing Company (1983), pp. 116-130. On p. 128 of my copy (corresponding to p. 239 of the IEEE paper, I presume) Jaynes, after deriving from a group-theoretic argument the prior θ ^{− 1}(1 − θ) ^{− 1}, remarks: "The prior (60) thus accounts for the kind of inductive inferences noted in the case of the chemical, which we all make intuitively. However, once we have seen at least one success and one failure, then we know that the experiment is a true binary one, in the sense of physical possibility, and from that point on all posterior distributions (69) remain normalized, permitting definite inferences about θ."

The reference to "the chemical" in this excerpt refers to Jaynes' example on the previous page, where he discusses a chemical dissolving or not dissolving in water, with the inference that it will do so reliably if it does so once; only when both cases are observed does one strongly think that the parameter might be on (0,1).

I infer from the passage that Jaynes would say that if we have one success and one failure, then for all other observations (excluding these two), the prior would be flat (after applying Bayes' theorem to these two observations using the prior displayed above).

Parenthetically, the Jeffreys prior, which many feel to be the right one in this case, is θ ^{− 1 / 2}(1 − θ) ^{− 1 / 2}--Billjefferys 18:50, 4 Apr 2005 (UTC)

I have expanded the section on uninformative priors and the references.--Bill Jefferys 18:53, 26 Apr 2005 (UTC)

[edit] Exponents reversed?

In the article, the Jeffrey's prior for a binomial proportion is given as p^{1 / 2}(1 − p)^{1 / 2}. However, a number of other sources on the internet give a Beta(0.5,0.5) distribution as the prior. But this corresponds to p ^{− 1 / 2}(1 − p) ^{− 1 / 2}. Similarly, my reading leads me to believe that the Jaynes' prior would be a Beta(2,2) distribution, corresponding to p¹(1 − p)¹, rather than the negative exponent.

Is it standard to give a prior in inverted form without the constants, or is there some convention I am unaware of? If so, perhaps it would be good to include it in the page. As a novice, I am puzzled by the introduction of, for example p ^{− 1}(1 − p) ^{− 1}, for which the integral over (0,1) doesn't even exist, as a prior, as well.--TheKro 12:50, 13 October 2006 (UTC)

You're right, the exponents should be − 1 / 2 in the case of the Jeffreys prior (note, no apostrophe, it's not Jeffrey's). The Jaynes prior is correct; it is an improper prior.

I have corrected the article. Bill Jefferys 14:41, 13 October 2006 (UTC)

p ^{− 1}(1 − p) ^{− 1} was not original to Jaynes, but that was the form he used. In general the normalizing constant is not vital, as explained in the first part of the improper priors section, since it is implicit for proper priors and meaningless for improper ones.--Henrygb 09:13, 14 October 2006 (UTC)

[edit] a priori

Hello all, I found this article after spending quite some time working on the article a priori (statistics). I'm thinking that article should probably be integrated into this one, what do people think? The general article on a priori and a priori (math modeling) are both in need of work, and I thought I would engage some editors from this article into the work. Really, the math modelling article should be integrated as well. jugander (t) 22:02, 14 October 2006 (UTC)