# 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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### Sufficient Statistic for Absolutely Continuous Distribution [duplicate]

The following is a homework problem. Please tell me if my solution is correct and if not please point out my mistakes. Let $x_{1}, x_{2},...,x_{M}$ be i.i.d. samples from the absolute continuous ...
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### Is a parameter-independent support sufficient to belong to an exponential family?

From Wikipedia (emphasis mine): As a counterexample if these conditions are relaxed, the family of uniform distributions (either discrete or continuous, with either or both bounds unknown) has a ...
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### What are the natural parameters of a matrix-normal distribution?

Assuming that the matrix-normal distribution is in the exponential family (as is a multivariate normal), what are its natural parameters?
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### Why is the canonical parameter linearly related to the input x in GLMs and why does it give the link function?

In Andrew Ng's CS229 notes, one of the three assumptions he makes for constructing GLM models is: The natural parameter $\eta$ and the inputs x are related linearly: $\eta=\theta^Tx$ He goes on to ...
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### How to determine sample space, $\sigma$-algebra and probability measure from the exponential family?

The sample space of binomial distribution is the set $\{0,1\}$ and its $\sigma$-algebra is the power set of $\{0,1\}$ while the sample space of normal distribution is $\mathbb R$ and its $\sigma$-...
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### Are These Conjectures Regarding Sufficient Statistics True?

I have these conjectures that I cannot quite prove (unless I impose another regularity condition of parameter-independent support for distribution, in which case, the conjectures are trivially true ---...
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### Technical term for natural statistics comined with log-partition

One way to find the posterior of an exponential family distribution with a conjugate prior is to use the natural reparametrization of the likelihood and prior and combining the sufficient statistics ...
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### Laplace distribution as an Exponential Distribution and Minimizitaion of KL Divergence

In the context of Expectation Propagation [Minka's thesis-2001], I would like to approximate an unknown distribution with a Laplace distribution. This can be solved by minimizing KL-Divergence. In ...
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### Practical method to do MLE for natural parameters in exponential family

I encountered the following question in my research and I hope this is the correct place to post it. I'm following the notation in this lecture note by Michael I. Jordan. Assume random vector $X$ ...
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### Why is the EM algorithm well suited for exponential families?

I've been brushing up on the EM algorithm, and while I feel like I understand the basics, I keep seeing the claim made (e.g. here, here, among several others) that EM works particularly well for ...
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### Example of curved exponential family with $T$ being a complete statistic?
Is there any example of curved exponential family with $T$ being a complete statistic? Here $T$ is the sufficient statistic.
### UMP test for exponential family when sufficient statistics $T$ is a vector
Assume we have a random sample $X_1,\dots,X_n$ from a distribution of the form $f(x_i;\theta) = h(x)g(\theta)\exp(\eta(\theta) T(x))$ and we wish to test \$H_0: \theta \leq \theta_0, H_1: \theta > \...