Talk:Exponential family

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Question: must the reference distribution H be a probability distribution, or will a positive measure do? Note that the normalization condition applies to F, not to H. I am thinking of cases where the reference "distribution" is Lebesgue measure (to wit: the Normal distribution) or counting measure. — Miguel 11:08, 2005 Apr 16 (UTC)

Certainly it is Lebesgue measure in some cases. And counting measure on positive integers -- clearly not assigning finite measure to the whole space -- in some cases. Which causes me to notice that this article is woefully deficient in examples. I'll be back.... Michael Hardy 20:36, 16 Apr 2005 (UTC)

[edit] Article reversion

Would you mind explaining the reversion of my edits on the article on the Exponential family? — Miguel 07:14, 2005 Apr 18 (UTC)

It said:

A is important in its own right, as it the cumulant-generating function of the probability distribution of the sufficient statistic T(X) when the distribution of X is H.

You changed it to:

A is important in its own right, as it is the cumulant-generating function of the probability distribution of the sufficient statistic T(X).

The edit consisted of deleting the words "when the distribution of X is H. The statement doesn't make sense without those words. A cumulant-generating function is always a cumulant-generating function of some particular probability distribution.

Well, actually the derivatives of A(η) evaluated at &eta instead of at zero give you the cumulants of dF(x|η), which is what I meant. The cumulants of dH are actually irrelevant to dF, what is interesting is that the cumulants of the entire family of exponential distributions with the same dH and T are encoded in A. Miguel 09:34, 2005 Apr 19 (UTC)

I see now that you also changed some other things. I haven't looked at those closely, but I now see that you changed "cdf" to "Lebesgue-Stieltjes integrator". I don't think that change makes sense either. That it is the Lebesgue-Stieltjes integrator is true, but the fact that it is the cdf is more to the point in this context. Michael Hardy 21:59, 18 Apr 2005 (UTC)

Except that, as you agree above, dH need not be a probability distribution, and hence it need not have a cdf. It is a positive measure, and dH is the integrated measure. I have never seen x on the whole real line called a cdf. It is fine if you want to call it a cdf, but then you'll have to explain somewhere else that the corresponding probability distribution may be "non-normalizable", and that will raise some eyebrows (not mine, though).

You also reverted a lot of valid content about the relationship between the exponential family and information entropy, as well as a reorganization of the existing information into sections, plus placeholders for discussing estimation and testing. The two edits that so bothered you were the last of a long series spanning two days. You could have been a little more careful. The appropriate thing would have been to discuss these things on this page. Miguel 09:34, 2005 Apr 19 (UTC)

[edit] Disputed

The current revision says that the Weibull distributions do not form an exponential family. This seems to ignore non-canonical exponential families (where the natural parameter may be transformed by another function). What am I missing? --MarkSweep (call me collect) 23:28, 14 November 2005 (UTC)

I may have lifted that from another source without checking the pdf. This is not open to interpretation. Either the Weibull distribution is exponential or it isn't according to the definition. Transforming the natural parameter is not the issue: the normal distribution's natural parameters are not the mean and variance. Miguel 09:31, 15 November 2005 (UTC)

AFAIK, it's not an exponential family according to the definition given in this article. So I guess I put the {{dubious}} tag in the wrong place. What I was trying to point out is that there is a more general definition of exponential family in common use, see e.g. [1]. The difference between these definition becomes apparent when one considers the Weibull distribution: it's not an exponential family according to the definition used here, but it is according to the more general definition (unless I'm missing something). --MarkSweep (call me collect) 19:49, 15 November 2005 (UTC)

If you "transform" the natural parameter, you're using a different parametrization of the family of probability distributions involved, but you're not looking at a different family of probability distributions. Is that what you're talking about? Michael Hardy 22:46, 15 November 2005 (UTC)

The Weibull distribution is not in the exponential family according to the definition given here, which is one that can also be found in widely-used textbooks such as Casella and Berger, Statistical Inference (2nd edition), 2002, page 114. Later in the article it is noted that you NEED this specific definition for a distribution to have sufficient statistics (a result of Darmois, Koopman and Pitman from the 1930s), so you can't generalize the definition any further without penalty. To shorten the controversy about Weibull, I propose that that ALL MENTION of Weibull be dropped from the article, leaving only Cauchy as the agreed-upon non-member of the exponential family. Ed 02:01, 16 June 2006 (UTC)

OK, further study does not find any references supporting Weibull in the exponential family, so I suggest that we remove the "dispute" tag and let the original wording stand, the one where Cauchy and Weibull are both left out of the family. I consulted books on the exponential family by Lawrence Brown and Ole Barndorff-Nielsen. EdJohnston 17:14, 23 June 2006 (UTC)

[edit] Question

Questions: Is the "prior" mentioned in the opening section the same as the "prior distribution" mentioned later? If so isn't the convention that the first mention of something links to the wikipedia article on it? (And calling it the "prior distribution there would be good too - people like me don't actually know any stats)

Also shouldn't "cdf" appear in brackets after the words "Cumulative distribution function" for clarity rather than just straight in the text? 20:25, 18 May 2006 (BST)