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)
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)
