# Tagged Questions

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### GLM (conditional/unconditional) distribution

Based on my readings about GLM, I am pretty sure that when we say the distribution of the response variable $y$ is a member of exponential family of distribution, what we really mean is that ...
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### Why computing P(x,D) is simpler than P(x|D) in exponential bayesian networks?

I am reading this tutorial on variational inference and wonder why the statement in the question title which is mentioned on page 3 is true.
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### Expected time between two events

I'm having trouble with the following problem: Consider a game between two players A and B. Player A must complete three tasks each of which take an exponentially distributed amount of time with ...
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### Implication of Theorem on 'Empirical Bayes optimality' (Morris 1983)

My question is about Theorem 1 from Morris 1983; Parametric Empirical Bayes Inference (http://www.jstor.org/stable/2287098?seq=1#page_scan_tab_contents). Suppose Y has a univariate natural ...
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### Karlin-Rubin theorem and exponential family

Let $X_1,...X_n$ a random sample of $X$ with density $$f(x;\alpha,\beta)=\frac{\alpha}{x^{\alpha+1}}\beta^\alpha I_{[\beta,\infty]}(x)\space\alpha>0,\beta>0$$ with $\beta$ known. Find a ...
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### Sufficient Statistic for non-exponential family distribution

Question: Let $X_1,X_2,....X_n$ be an iid sample from $N(\theta , 4 \theta^2 )$. I want to show that this model is not a member of the exponential family and to find a sufficient statistic for $\theta$...
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### normalization constant for categorical distribution as exponential family

Let r.v. $X$ has categorical distribution. We can represent its pmf as $f(x\mid\vec{p})=\Pi_{i=1}^{K}p_i^{I[x=i]}=\exp[\sum_{i=1}^{K}I[x=i]\ln p_i]$, there is no explicit normalization constant (...
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### Equivalent of z-score in exponential distribution

I have an exponentially distributed set of data, but there are a number of outliers at the margin. If I want to identify samples above the mean that I am 99.9+% confident are not the result of ...
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### Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional sufficient ...
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### Role of base measure in exponential family

An exponential family distribution $p$ in the canonical form can be written as $p(x|\theta) = h(x)\exp(\theta^\top T(x) - A(\theta))$ where $A(\theta)$ is the log partition function, $T(x)$ is the ...
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### Log Likelihoods of Exponential Families

How can one derive the log-likelihood of the saturated model of an exponential family in general? Differentiating the log likelihood w.r.t $\theta$ gives $y_i=\hat{\mu_i}$ but I don't think replacing ...