# Questions tagged [exponential-family]

A set of distributions (eg, normal, $\chi^2$, Poisson, etc) that share a specific form. Many of the distributions in the exponential family are standard, workhorse distributions in statistics, w/ convenient statistical properties.

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### Why is the exponential family so important in statistics?

Why is the exponential family so important in statistics? I was recently reading about the exponential family within statistics. As far as I understand, the exponential family refers to any ...
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### When if ever is a median statistic a sufficient statistic?

I came across a casual remark on The Chemical Statistician that a sample median could often be a choice for a sufficient statistic but, besides the obvious case of one or two observations where it ...
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### Does a canonical link function always exist for a Generalized Linear Model (GLM)?

In GLM, assuming a scalar $Y$ and $\theta$ for the underlying distribution with p.d.f. $$f_Y(y | \theta, \tau) = h(y,\tau) \exp{\left(\frac{\theta y - A(\theta)}{d(\tau)} \right)}$$ It can be shown ...
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### Exponential Family: Observed vs. Expected Sufficient Statistics

My question arises from reading reading Minka's "Estimating a Dirichlet Distribution", which states the following without proof in the context of deriving a maximum-likelihood estimator for a ...
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### Conditional distribution for Exponential family

We have a random variable $X$ that belongs to the exponential family with p.d.f. $$P_X(x|\boldsymbol \theta) = h(x) \exp\left(\eta({\boldsymbol \theta}) . T(x) - A({\boldsymbol \theta}) \right)$$ ...
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### Why do we assume the exponential family in the GLM context?

When I first learned about Generalized Linear Models I thought that the assumption that the dependent variable follows some distribution from the exponential family was made to simplify calculations. ...
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### Role of base measure in exponential family

An exponential family distribution $p$ in the canonical form can be written as $p(x|\theta) = h(x)\exp(\theta^\top T(x) - A(\theta))$ where $A(\theta)$ is the log partition function, $T(x)$ is the ...
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### Support vector machines (SVMs) are the zero temperature limit of logistic regression?

I was had a quick discussion recently with a knowledgeable friend who mentioned that SVMs are the zero temperature limit of logistic regression. The rationale involved marginal polytopes and fenchel ...
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### Distribution of sum of independent exponentials with random number of summands

Let $\tau_i\sim\exp\left(\lambda\right)$ be independent and identically distributed exponentials with parameter $\lambda$. Then, for given $n$, the sum of these values $$T_n := \sum_{i=0}^n \tau_i$$ ...
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### Is the negative exponential distribution a member of the exponential family?

Please correct me if I am wrong. The general form of $k$-parameter exponential family is $$f(x;\boldsymbol{\theta}) = a(\boldsymbol{\theta})g(x) \exp\{\sum_{i=1}^{k}b(\boldsymbol{\theta}) R_i(x)\}$$ ...
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### How do I find the UMVUE of $\sqrt{\alpha}$ here?

new user here self-studying some mathematical statistics. I came across this problem and am stuck. Problem: Suppose that for $i = 1, ... , n$, the positive random variables $X_i$ are independent and ...
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### Exponential family form of multinomial distribution

I feel like this must be a duplicate, but I don't know the magic words to find the appropriate post... The multinomial distribution is a member of the exponential family. I am used to seeing the "...
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### Intuition for why the (log) partition function matters?

I'm on a quest for the intuition behind the fact that theoretical introductions to approximate inference focus so much on the log partition function. Say we have a regular exponential family p(\...
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I am struggling to understand the following result from Casella and Berger about sufficiency and completeness for exponential families: Let $X_{1},X_{2},...,X_{n}$ be iid observations from an ...