# Questions tagged [conjugate-prior]

A prior distribution in Bayesian statistics that is such that, when combined with the likelihood, the resulting posterior is from the same family of distributions.

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### Using conjugate priors to estimate the posterior distribution of a proportion of a region composed by subregions

Let's say I have a region that is divided into 3 subregions. In each subregion, I run ~90-110 randomly allocated surveys asking a binary question. I want to know if the way that I am estimating the ...
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### Write the PDF of an exponential prior given E[$\theta$] = 2

I am reviewing old exercise solutions and the following info is given: Assume that the conjugate prior for θ (as a special case of the gamma distribution) is following the exponential distribution ...
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### Non-Dirichlet Prior for $Cat(\theta)$ parameter that can tractably be integrated out (for Latent Dirichlet Analysis)?

In LDA Topic Models, it is standard to 'integrate out' the $\theta$ parameter, which contains a document's Categorical probabilities of drawing each topic. QUESTION If one uses the standard Dirichlet ...
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### Shifted Inverse Gamma still conjugate

Suppose '''X~Inverse Gamma(a,b)''' over '''[0,\infty]''' and we have '''X+c''' over '''[c,\infty]''' for '''c>0''' known constant. The likelihood is normal with unknown variance, and the variance ...
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### Bayesian Poisson Regression with Gamma Prior Formulas

Are there closed form formulas for the posterior and evidence of a Poisson-Gamma Bayesian regression model? I was not able to find anything that is accessible online. I am not sure for which model can ...
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### The PDF of the Data (Marginal Likelihood) Given the Prior of a Gamma Distribution with Prior on the $\beta$ Paraneter

Given a model where $x_i | \beta \sim \mathcal{Gamma} ( \alpha, \beta )$ where $\beta \sim \mathcal{Gamma} ( \alpha0, \beta0 )$, is there a closed form formula for the PDF of $x_i$? Namely, what'...
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### Poisson-Gamma Model for Landslides

I am looking to derive the mean return time and relative likelihood for the following problem however the answer from a colleague comes up as 68?: We observe 3 landslide events occurring over 170000 ...
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### Find Bayesian estimator for $e^{\theta}$

Given $\{Y_i\}_n\sim U(\theta-1,\theta+1)$ and prior distribution $\theta\sim U(a,b),1\leq a<b$ is the posterior distribution conjugate? Find the absolute error estimator for $e^{\theta}$ and ...
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### Posterior derivation of normal model

Working through the book Bayesian Essentials with R by Jean-Michel Marin & Christian Robert I am trying to work out the posterior for the model given on page 29 when the data is from a normal with ...
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### Does a sufficient statistic imply the existence of a conjugate prior?

In the comments on this answer, user Scortchi asks: So iff there's a sufficient statistic of constant dimension, there's a conjugate prior? As far as I know this didn't get a complete answer, so I'm ...
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