Talk:Exponential family

message: 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)
thread_title:  Must H be a pdf 

message: 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)
replies: text: AFAIK, it's not an exponential family according to the definition given in this article.  So I guess I put the  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. .  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)

text: It is an exponential distribution and is one according to the definition given in this article. The cdf defines a measure. It does not depend on the particular parameterization of the family of distributions. The natural parameters of the Weibull are (λ^-k,k-1) where λ and k are the parameters given in Wikipedia's defn of the Weibull. H is then Lebesque measure, and you can work out A yourself. Note that the Weibull is a generalization of the exponential distribution which is Weibull with k=1. CWoo on Planetmath gets this right (though his article seems to simplify some things). Odometer 05:34, 27 December 2006 (UTC)
replies: text: To put Weibull in the exponential family needs a reference, in my opinion. Membership in the family is not expected to be invariant under transformation of the parameters. I left a question about this at User_talk:Odometer but have not received a response. I found that I was unable to 'work out A myself'. PlanetMath is interesting but it is not a reliable source for our purposes. EdJohnston 18:06, 8 January 2007 (UTC)

text: Look at the planetmath definition of exponential family. It's an equivalent definition when density functions exist, and it's the one that's often used because it's much easier to comprehend and pattern match on, so it's probably the more appropriate definition to use on wikipedia. Casella and Berger use that defn in their book. You just factor the density into a parameter part, a "data" part, and an interaction part. Generally speaking the 'A' doesn't matter that much when determining if it's an exponential family. 'A' is just the normalizing constant to make the density integrate to 1. It's sometimes called the log partition function while the rest of the density is sometimes called the kernel. In this case it's -k log(λ) + log(k) using the parameterization in wikipedia. You can stick that in the natural parameterization if you want. Also membership in the family is definitely invariant under transformation of the parameters. A family of distributions is just a collection of probability measures which are indexed by some parameters. You can change the indices, and it's still the same collection. Odometer 00:33, 16 February 2007 (UTC)
replies: text: Can you provide the functions a, b, c and d needed at  for the Weibull distribution? I would be happy if a simpler pattern for the exponential family could be used in this article. It's not crystal clear that measure theory is essential for explaining this stuff. It does set a high prerequisite for understanding the article.  EdJohnston 02:15, 16 February 2007 (UTC)
thread_title: Disputed

message: I just added a technical(expert) tag to this page. I understand and have used exponential family forms of distributions many times, but I find this article extremely difficult to read and almost willfully obscure. I can understand why one might want to use Lebesgue-Stieltjes integration, but the examples I have read have managed to avoid the topic completely and still dealt with continuous as well as discrete probability distributions. See for example, McCullagh and Nelder, "Generalize Linear Models" pp 28, or Givens and Hoeting, "Computational Statistics" pp 5. A read though the mathematics Wikiproject page, specifically, the section titled, "Some issues to think about" and the Wikipedia page on making technical articles accessible.
The general idea is to start simple (i.e. without invoking complicated concepts), which is definitely possible as I have noted two examples from textbooks, and then to get to more rigorous treatments later on. Thanks in advance to anyone who helps with this! O18 17:17, 1 June 2007 (UTC)
replies: text: Yes, this article would be more useful without the measure theory and the Lebesgue-Stieltjes integration. No books on my shelf seem to do a patient exposition of the exponential family. Maybe the treatment you found in McCullagh and Nelder would provide inspiration for our article? Also there's quite a bit of matrix notation used in the article and that adds to the reader's burden. EdJohnston 04:31, 24 June 2007 (UTC)
replies: text: I think the MC&N method is too focused on GLM, I think others might be better. O18 14:43, 24 June 2007 (UTC)
thread_title:  Technical(expert) tag 

message: The definition of the exponential family is given at the top of the page by dF(x|\eta) = e^{-\eta^{\top} T(x) - A(\eta)}\, dH(x), which has a minus sign before the first \eta.  However, section "Differential identities: an example" claims the "exponential family with canonical parameter" is not preceded by a minus sign.  Is the minus absorbed into \eta in the "Differential identities" in order for the Expectation and Variance formulas to work?  If so, perhaps the form of the exponential family should remain the same and this result be formulated consistent with the single form.
Also, section "Differential identities: an example" claims that "As mentioned above \scriptstyle K(u) = A(u + \eta) - A(\eta) ", but \scriptstyle K(u) is not previously mentioned.
Erik Barry Erhardt 21:10, 23 June 2007 (UTC)
replies: text: I personally prefer the definition with the plus sign, but then there is an inconsistency among the three definitions given at the beginning. The elementary single-parameter definition and the vector-parameter definitions have a + sign in front of the η but a - sign in front of the 'A', whereas the measure-theoretic definition has a minus signs on all terms. Miguel (talk) 17:17, 14 April 2010 (UTC)
thread_title:  minus sign mismatch? and no previous mention of <math>K(u)</math> 

message: This used to say that the uniform distribution is not in the exponential family.  But the uniform dist is a special kind of beta dist, and the beta dist is in the family. Benwing 06:02, 22 August 2007 (UTC)
replies: text: I've never seen anyone miss the point of a definition so completely.  Yes the uniform distribution is a beta distribution, yes, the beta distributions form an exponential family.  To suggest that that in some way implies that the uniform distributions form an exponential family is ridiculous.  You really need to read and understand the definition of "exponential family" before you can understand such things.  An exponential family is a not just a distribution; it is a family of distributions.  Every uniform distribution belongs to some exponential family, and in fact to more than one exponential family.  That does not mean that the family of distributions that includes only the uniform distributions is an exponential family.  It obviously is not, since the support of the various distributions in the family varies. Michael Hardy 00:16, 23 August 2007 (UTC)
replies: text:  OK, fine, I made a mistake, but you have too -- you've forgotten to be civil, see WP:CIVIL. Benwing 07:40, 23 August 2007 (UTC)
thread_title:  uniform dist 

message: Several editors have written an alternative version of this page that is intended to be more readable for readers with out Ph.D.s in maths. This effort  resulted form the above conversation, and is located at (User:Pdbailey/Sandbox/Exponential_family). I intend to replace the body of this article in a few days, if there are no objections. Please also feel free to edit the linked page before it is moved or after it is moved. O18 19:51, 22 September 2007 (UTC)
replies: text: Update: The rewrite has now been carried out. We hope you like it. Feel free to directly edit the exponential family page. --Zvika 08:38, 27 September 2007 (UTC)
thread_title:  proposed overhaul 

message: Hello, the article says that "In the case of a likelihood which belongs to the exponential family there exists a conjugate prior, which is often also in the exponential family." This suggests that exponential families exist with conjugate priors that are not in the exponential family, is that true? But the form given for the conjugate prior, \pi(\eta) \propto \exp(-\eta^{\top} \alpha - \beta\, A(\eta)), looks like it is in the exponential family (for instance if we append to the vector \eta a component A(\eta)). Thanks in advance. A5 10:05, 1 November 2007 (UTC)
thread_title:  Conjugate priors 

message: Pitman-Koopman theorem redirects to here; it deserves its own page.  The theorem is also not listed on this page under its own name.  Agalmic (talk) 01:05, 4 July 2008 (UTC)
thread_title:  Redirection from Pitman-Koopman theorem 

message: Mention of the negative binomial should be qualified, too. As far as I can tell, it wouldn't be in the an exponential family unless the parameter, r, is fixed. Schomerus (talk) 23:29, 12 October 2009 (UTC)
replies: text: I think you're right, and I changed the article accordingly. P.S. In the future please add new discussions at the bottom of the talk page. --Zvika (talk) 05:45, 13 October 2009 (UTC)
thread_title: Negative Binomial

message: I believe the exposition would be clearer and easier to understand if we start with the exponential family in its canonical form, where θ is the natural parameter of the family. In that way we would be able to talk about the dual parameters, and about estimation of the parameters (e.g. the theorem that the exponential family is the only family where parameters can be estimated efficiently).  … stpasha »  19:34, 24 December 2009 (UTC)
thread_title:  Canonical parameters 

message: I edited
: Suppose H is a non-decreasing function of a real variable and H(x) approaches 0 as x approaches −∞.  
into
: Suppose H is a non-decreasing function of a real variable.
Clearly we want to include the case where H(x) = x. — Miguel (talk) 16:53, 14 April 2010 (UTC)
thread_title:  Measure-theoretic formulation excludes dx ! 

message: How do we read the expressions fx(x|θ) and fσ(x;μ)? While I'm on the topic, wouldn't it be better if every new symbol introduced into a mathematics article were also posted in the Table of Mathematical Symbols []? As it is, whenever I read a mathematical article, I am then forced to look up the symbols elsewhere to get the pronunciation, rendering the otherwise fine math content of Wikipedia pretty much redundunt. Wikipedia is, after all, supposed to be a resource for the non-expert as well as the expert.  — Preceding unsigned comment added by Romanicus (talk • contribs) 17:14, 16 December 2010 (UTC)
replies: text: I guess you are looking for the Greek alphabet article?  // stpasha »  21:41, 16 December 2010 (UTC)

text: They're not read, they're written. Higher-level maths isn't a spoken language, it's a written one. Even when face-to-face,  mathematicians need pencil and paper, or blackboard and chalk, or whiteboard and marker, to communicate. --Qwfp (talk) 08:57, 15 January 2011 (UTC)

text: Since x can be discrete, differentiability of h(x) can play no role. Melcombe (talk) 12:54, 16 March 2011 (UTC)
thread_title:  Newbie question. . . but it really bugs me! Math Notation, read as . . .

message: The MGF given is stated to apply for all exponential families...  I believe this might be an error and that it only applies to Natural Exponential Families.  If someone can refute this please do so.  Otherwise I will try to find a reference that clears up the matter one way or the other....  —Preceding unsigned comment added by 137.194.233.21 (talk) 12:41, 22 April 2011 (UTC)
thread_title: Query on Correctness of Moment generating function section

message: In the subsection "scalar parameter", there is:
:More generally, η(θ) and T(x) can each be vector-valued such that \eta(\theta)'\cdot T(x) is real-valued.
What is the meaning of  \eta(\theta)'\cdot T(x)?
Is it supposed to be \eta(\theta)^T T(x)?
--Bongilles (talk) 15:54, 13 April 2016 (UTC)
thread_title:  What is the meaning of the prime notation here: <math>\eta(\theta)'\cdot T(x)</math> 

message: The current statement of the Pitman-Koopman-Darmois theorem seems to be missing some regularity conditions. The latest development I am aware of is the proof by Barndorff-Nielsen and Pederson (1968) in the paper "Sufficient data reduction and exponential families." There, beyond the "support does not change across the family" assumption, in all cases it is required that the probability densities for the family are continuous and that the statistic is continuous. Moreover, when the dimension of the sufficient statistic is greater than 1, it also is required that the densities have continuous partial derivatives.
I can make a (suitably simple-to-read) version including the appropriate regularity conditions, but before that I wanted to see if anyone knows if any of the regularity conditions can be relaxed; i.e. are there any results beyond those from Barndorff-Nielsen and Pederson?  Tekhnofiend (talk) 21:29, 15 May 2016 (UTC)
thread_title:  Regularity conditions for Pitman-Koopman-Darmois theorem 

message: This is far from complete listing. The most useful information is the log-partition function. 68.134.243.51 (talk) 20:37, 13 September 2022 (UTC)
thread_title:  Table of distributions needs more distributions 

message: The section Factorization of the variables involved appears to be original research. It has no citations, and it is not very helpful. It seems to be an inconclusive attempt by an editor to provide a recipe for recognizing when an expression can be put into the form of an exponential family. 
It is abstract, but not complete. It is a rule of thumb that immediately has exceptions, not a systematic method.
It would be more useful to replace this section by some examples. Or rewrite it so that it does not make a claim to comprehensiveness.
2A02:1210:2642:4A00:5846:AA29:F37C:805C (talk) 11:57, 23 June 2023 (UTC)
thread_title:  Section entitled "Factorization of the variables involved" 

message: Biggerj1 (talk) 18:48, 29 October 2023 (UTC)
thread_title:  Is the [[Tweedie distribution]] family an exponential family in the sense of this article?