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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How do I find the $Var(t(x))$ of an exponential family?

Suppose $X$ is a random variable whose distributions form an exponential family, with pmf or pdf given by $f(x|\theta)=h(x)c(\theta)exp\{w(\theta)t(x)\}.$ How can I find the $Var(t(x))$? I have seen ...
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expectation of conditional categorical distribution

I have the following network with $x$ and $y$ both binary: I've written $p(y|x)$ in exponential family form as: Now I need to get the expectation of this distribution, but I'm not sure how to ...
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Ambiguity with math notation

I was just solving several problems and get stuck into the one with some weird notation. I wasn't able to understand it, though I know the meaning behind the character (angle bracket) in more broader ...
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Moment parametrization for exponential families where the observation is not a sufficient statistics

Consider a probability distribution belonging to an exponential family $$f(y; \theta) = c(y) \exp ( \eta(\theta)^\top T(y) - \kappa(\theta))$$ In the case where $\eta(\theta) = \theta$ (always ...
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Expectation of exponential family distributions

Is there a closed form of the following marginal (one dimensional data) $\pi(\theta|y) = \mathbb{E}_{x \sim \pi_R(x|y)} \pi(\theta|x)$, where both $\pi, \pi_R$ are exponential family distributions?
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If $f(x|\theta)$ is conjugate to $p(\theta)$ then is $f(x|r\theta)$ conjugate to $p(\theta)$?

If exponential family $f(x|\theta)$ is conjugate to $p(\theta)$ then is $f(x|r\theta)$ for $r>0$ conjugate to $p(\theta)$? If not, what can we do about it in terms of sampling to make use of ...
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Percentages in utilization of different distributions

I'm sure that this question has been asked before on CV but, in drilling through many pages of previous CV questions, no matches surfaced. Regardless, I'm confident some observant participant will be ...
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Role of different exponential families in generalized linear models

I've gone through a variety of introductions to generalized linear models (GLM), and there's always a point in the discussion that confuses me. The story often begins saying that $P(y|x)$ belongs to a ...
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Proof that one distribution is a GLM (general linear model)

Given $Y_1, Y_2,...,Y_n$ i.i.d random variables, where $Y_i|x_i \sim N(\mu_i, \sigma^2)$ and $\mu_i = \beta_0 + \log(\beta_1 + \beta_2x_i)$. How do I proof that the distribution is a GLM (general ...
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Mixed parameterization of sample from normal distribution

I am studying exponential families and mixed parameterizations. Now, I am told that $$\mathbf{\theta} = \begin{bmatrix}\mu\\ -\frac{1}{2\sigma^2}\end{bmatrix}$$ is the parameter in a variation-...
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obtain (minimal) sufficient statistic for $\gamma$ knowing the canonical statistic $\theta(\gamma)$

A statistical model for a data set y is an exponential family , with canonical parameter vector $\theta= (\theta_1,\theta_2,..\theta_k)$ and canonical statistic $t(y) = (t_1(y),t_2(y),..t_k(y))$ if ...
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UMVUE for $g(p) = \mathbb{E}_p[X^2]$, where X follows a geometric distribution

I have a random variable X with pmf $$p_\lambda(x) = (1-p)^{x-1}p, \ \ x = 1,2,3,\ldots, \ \ p \in (0,1)$$ and I am trying to find a UMVUE for $$g(p) = \mathbb{E}_p[X^2]$$. Here is my attempt so ...
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Find a UMP test in exponential family - first stage

Consider the exponential family $f(x;\theta) = c(\theta)\exp\{b(\theta) T(x))\}h(x)$ and I would like to find a UMP test for $$H_0: \theta \leq 0\quad \hbox{vs}\quad H_1: \theta > 0.$$ It is ...
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Is the marginal distribution of a maximum entropy distribution also a maximum entropy distribution?

I am considering under what circumstances maximum entropy distributions are closed under marginalization. The main case I am interested is the following setting: a finite set of discrete random ...
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Sufficient Statistic and MLE

Suppose $X_1, \dots, X_n \sim B(1,p)$. Show that a sufficient statistic for $\theta = (1-p)^2$ is $T(x) = \sum X_i$ and that the MLE for $\theta$ is $(1-\frac{1}{n}T)^2$. I am having a lot of ...
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What is matrix norm in (Collins, 2001)

I am studying "A generalization of PCA to the exponential family" (Collins et al., 2001) and I don't understand some notations. What is the meaning of the matrix squared norm on page 6 ? Is it a ...
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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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Gamma distribution as a member of exponential family

In my lecture notes I have that the distribution of a random variable $Y$ is said to be in the exponential family if it can be written as $f(y;\theta)=exp(a(y)b(\theta)+c(\theta)+d(y))$, where $a,b,c$ ...
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Finding the UMVUE of $\theta^2$ where $f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of ${\theta^2}$ I ...