# 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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### Distribution of sampling without replacement

Consider $N$ items with associated weights $w_i$. Each time, we sample one item from the remainder without replacement and the sampling probability is proportional to the weights. Continue sampling ...
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### Fitting a GLM – whether to estimate mean or dispersion first

I'm trying to understand how fitting GLMs works. I asked before on math.SE but this community is probably more appropriate. My confusion is the following: The MLE for $\hat\mu$ uses $\phi$. My books ...
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### KL Divergence, Bregman, and uniqueness

While reading the following paper on Bregman Divergence (link) Banerjee, Arindam, et al. "Clustering with Bregman divergences." Journal of machine learning research 6.Oct (2005): 1705-1749. In ...
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### How can variance be non constant for Bernoulli and Poisson

Until now I had learned that the variances of Bernoulli and Poisson random variables are $p(1-p)$ and $λ$ and that for fixed $p$ and $λ$, these variances are constant. Now, introducing glm, my course ...
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### What is sufficient statistics in the context of generalized linear models

the three assumptions Andrew Ng makes when deriving GLM are: $y\mid x;\theta \sim \operatorname{ExponentialFamily}(\eta)$ Our goal is to predict the expected value of $T(y)$ given $x$. Since in most ...
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### Gaussian Exponential Sum Approximation

Let $x_j,\ j\in\{1,\cdots,n\}$ be independent standard Gaussian random variables and $a_j,\ j\in\{1,\cdots,n\}$ be constants. What would a function $f$ and a standard Gaussian random variable $x_0$ ...
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### Formula for the covariance of logs in exponential families

I could not understand and don't see how the author derived equation (16). The Covariance is with respect to a pdf $\mu_k$ for the random vector $u$. The variable $\xi$ is a ratio of two densities and ...
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### What r parameter is used in a negative binomial regression?

From my understanding of the negative binomial regression, we have $Y_i|X_i; \theta$ is distributed $Neg Binom (r_i, p_i)$, where $r_i$ is known and fixed (analogous to a fixed $\sigma^2$ when we ...
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### Help Deriving Variance Function - Binomial GLM

I'm having difficulty replicating/deriving a result in GLM's for Binomial data. That is, if $Y \sim Bin(n, \mu)$ and we put the distribution of $Y/n$ into exponential family form (with a dispersion ...
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### Are there any non exponential family distributions with conjugate priors? [duplicate]

I believed I had been taught that only exponential family distributions have conjugate priors but I have recently read that ' all exponential family distributions have conjugate priors', leaving the ...
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### Zero-truncated Poisson model

In the theory of generalised linear models, you may use the exponential family to find the mean and variance of certain distributions. How would the mean and expectation of the zero-truncated Poisson ...
I recently saw* a pmf: $f(y)=\frac{\mu^y}{(y!)^\theta z(\mu,\theta)}$, where $z(\mu,\theta) = \sum_{i=0}^{\infty}\frac{\mu^i}{(i!)^\theta}$. * It is a bonus question on a homework assignment. My ...
I want to understand how the variance of an exponential family behaves. To take a very concrete example. Let consider the unit ball $B$ in d dimensions. Consider the following distribution over unit ...