# 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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### Can the Beta-regression be written in the GLM form?

The Beta distribution is: $$p(y)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}y^{\alpha-1}(1-y)^{\beta-1}$$ It's part of the exponential family. We can reparametrize this with using mean ...
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### Calculating the exponential growth rate against the standard deviation of the year coefficient

I have time-series abundance data for various locations. I would like to calculate the exponential growth rate for each location against the standard deviation of the year coefficient. My dataframe ...
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### Why does $E\left(y_{i t}\right)=a^{\prime}\left(\theta_{i t}\right)$? in the context of assuming some GEE marginal density?

In generalized estimating equations we have a glm-response variable. To establish notation, we let $Y_{i}=\left(y_{i 1}, \ldots, y_{i n_{i}}\right)^{\text {T }}$ be the $n_{i} \times 1$ vector of ...
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### Prove sum of T(Xi) also belong to exponential family

In mathematical statistics, suppose we have an independent and identically distributed sample X = ($X_1$, $X_2$, …, $X_n$) from a distribution that belongs to the exponential family. Say we can write ...
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### Inhowfar does Variational Inference work better with members of the exponential family?

I am reading Variational Inference: A Review for Statisticians. Working in [the exponential] family simplifies variational inference: it is easier to derive the corresponding CAVI algorithm, and it ...
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### What is the dispersion parameter of binomial distribution?

Binomial distribution is a member of exponential dispersion models, but I can not find the dispersion parameter of it. Could anyone help me find it out? IMO the Binomial distribution only has an ...
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### Is there a special case of the EM algorithm for exponential family distributions?

According to Wikipedia, the formal definition of the EM algorithm is The EM algorithm seeks to find the MLE of the marginal likelihood by iteratively applying these two steps: Expectation step (E ...
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### Fit exponential decay upwards model - start values give 'convergence failure'

I have some data that when plotted looks similar to this: Then eyeballing the charts on this page, it looks like I might have an exponential decay (Increasing form) relationship? I searched for how ...
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### Entropy of Poisson random variable via exponential family identity

Exercise 8.1 of Probabilistic Graphical Models asks the reader to use the identity $$H_{P_{\theta}}(X)= \ln Z(\theta) - \langle E_{P_{\theta}}[\tau(X)], t(\theta)\rangle$$ to compute the entropy $H$ ...
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### Asymptotic normality of MLE

We know under regularity conditions the MLE is asymptotically normal. Usually, it is said that in practice it's hard to check these assumptions. However, I wondered whether we can claim that these ...
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Let $X_1, \ldots, X_n$ be a sample from an exponential distribution with p.d.f. $f(x; \theta) = \theta e^{-\theta x}$ for $x > 0$ where $\theta > 0$ is an unknown parameter. I would like to find ...