Talk:Natural exponential family

What is the definition?

It is not clear from this article what the difference between EFs and NEFs is. Are the properties specified here a definition, or are they a result of the definition? Are some of the properties sufficient to guarantee that the others hold too? --Zvika 06:16, 18 October 2007 (UTC)

It's also no clear how there is a single parameter normal. It appears to imply (though it obviously can't) that the normal has one parameter if you think of it this way. Pdbailey 02:33, 20 October 2007 (UTC)

Because of the above two claims, and general disclarity on this page, I've added a cleanup tag. Pdbailey 02:34, 20 October 2007 (UTC)

The normal is single parameter if you consider the variance to be fixed rather than estimated. But there are other unclear things including the notation: for instance why T-dot not just T ? If someone could make the notation consistent and clarify the relationships between here, exponential family and Generalized_linear_model#Distribution_Function that would be wonderful (if they could also look at Tweedie distributions that would be amazing...) Qwfp (talk) 12:36, 23 January 2008 (UTC)

I've rewritten most of it, hopefully it should be clearer, though it could do with some citations. The bit at the end could do with a rewrite as well. -81.157.199.65 (talk) 17:14, 30 March 2008 (UTC)

Yes, that was extremely helpful, thanks. Pdbailey (talk) 20:19, 30 March 2008 (UTC)

Distributions

So, given the present definition, what would be the parameterization for the normal distribution? Pdbailey (talk) 00:20, 31 March 2008 (UTC)

Just got it, sorry. Pdbailey (talk) 00:21, 31 March 2008 (UTC)

I tried to do the exponential and gamma, but it looks like you end up taking the log of a negative number when you do that. Or am I missing something? BTW, if you don't like the table, feel free to just remove it, I'm not sure it is a great idea. Pdbailey (talk) 00:58, 31 March 2008 (UTC)

The key is to remember the parameter space changes. If β is the usual inverse-mean parameter, it has parameter space $\beta \in (0,\infty )$. The parameter space for the natural parameter in the NEF is $\theta =-\beta \in (-\infty ,0)$, thus it is possible to take $\log -\theta$. As for the table, personally I don't know if it adds a lot to the article in terms of content (and purely from an aesthetic point of view, tables containing formulae are pretty ugly). It may be better to have one or two fully worked examples (e.g. Poisson and normal) to explain the concepts -81.157.199.65 (talk) 14:09, 31 March 2008 (UTC)

Thanks! Your idea sounds good to me, but the normal is already on the exponential family page in NEF. How about the Poisson and the Gamma. Pdbailey (talk) 17:01, 31 March 2008 (UTC)

Question

The article states that convolutions of iid NEFs are also NEFs... Is it also true that the convoltion of NEFs with different parameter values are also NEFs... i.e. is the independant criterion required?... —Preceding unsigned comment added by 137.194.233.21 (talk) 15:00, 21 April 2011 (UTC)

Relation to Exponential Tilting

The Natural exponential family seems to be equivalent to the Exponential Tilting method and Esscher transform. Would it make sense to merge the articles? I'm not sure which one should then be the root. — Preceding unsigned comment added by Thomasda (talk • contribs) 11:37, 1 March 2021 (UTC)