# 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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### 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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### How to derive the conjugate prior of an exponential family distribution

I am trying to derive the conjugate prior of the univariate Gaussian distribution over both the mean and the precision. I know that the prior I'm looking for is the normal-gamma distribution, but the ...
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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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### Find UMVUE of $\frac{1}{\theta}$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of $\frac{1}{\theta}$ ...
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### Unbiased estimator with minimum variance for $1/\theta$

Let$X_1, ...,X_n$ be a random sample feom a distribution $Geometric(\theta)$ for $0<\theta<1$. I.e, $$p_{\theta}(x)=\theta(1-\theta)^{x-1} I_{\{1,2,...\}}(x)$$ Find the unbiased estimator ...
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### What is meant by the term 'exponential family'? Why it is named so?

I have come across the term exponential family. The Bernoulli, Gaussian and many more distributions come under this exponential family. What would be the commonalities between them?
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### Why is the mixtures of conjugate priors important?

I have questions about the mixture of conjugate priors. I learned and saw the mixture of conjugate priors a couple of times when I am learning bayesian. I am wondering why this theorem is such ...
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### Is the negative binomial not expressible as in the exponential family if there are 2 unknowns?

I had a homework assignment to express the negative binomial distribution as an exponential family of distributions given that the dispersion parameter was a known constant. This was fairly easy, but ...
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### Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$

Suppose $X_1, X_2, . . . , X_n$ are i.i.d Poisson ($\theta$) random variables, where $\theta\in(0,\infty)$. Give the UMVUE of $\theta e^{-\theta}$ I found a similar problem here. I have that the ...
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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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### 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 ...