# 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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### Conditionally conjugate prior for non-nested (i.e. crossed) normal model?

I am trying to write/understand a conditionally-conjugate Gibbs sampler for what is essentially a linear, mixed effects model. I more or less get the conditionally-conjugate posterior for the ...
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### Equivalent of Conjugate Priors for Marginal Probability Distributions

In probability, there are nice “conjugate prior” distributions that enable closed-form Bayesian updating – e.g. if you have a Normal likelihood and Normal prior (on the mean parameter), you get a ...
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### How to calculate the posterior distribution with a normal likelihood function and a prior that involves sigma

In the problem, the data X follows a normal distribution, or $f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{1}{2}(\frac{x-\mu}{\sigma})^2)$. Let's say I know the value of $\sigma^2$ and ...
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### Eliciting a Gamma informative prior in a Gamma–Poisson Bayesian problem

I employ the Gamma–Poisson conjugate family for my statistical model. I want to use an informative prior. From theory, I know that the values of the Gamma-distributed random variable lie within the ...
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### Pareto distribution with Gamma prior on parameter $\theta$

I want to calculate the posterior distribution of Pareto distribution with known parameter $X_m$ and unknown parameter $\theta$, with conjugate prior on $\theta$ the Gamma distribution: My effort is ...
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### Necessity of Metropolis Hastings algorithm for given posterior distribution

Let's say that we have calculated the posterior distribution of a parameter of interest given the data of a binomial experiment $N=70,x=34$ which the probability of event occurrence $\theta$ follows ...
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### Why not use the same distribution for the prior in Bayesian statistics?

I am wondering why introductory books on statistics use a conjugate distribution family for the prior instead of using the same pdf of the one we are trying to infer the parameters? For example, the ...
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### In what ways do conjugate priors compose?

A lot of conjugate priors are known for a lot of likelihood distributions (mostly the exponential family). But most Bayesian models in practice don't just consist of one distribution. Usually, you ...
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### Exponential family and conjugate priors

Is a distribution that belongs to the exponential family necessarily conjugate prior?
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### Can a prior be conjugate and noninformative at the same time?

And if so, could somebody give me a concrete example?
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### Interpretation of "virtual observations" in Bayesian inference

In the context of using a Normal-Gamma conjugate prior, scale $\eta_0$ can be interpreted as "virtual observations" which is a multiple of variance. Why then is the higher the "virtual ...
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### Bayesian inference - Calculating the prior distribution of the parameter in the Bernouli distribution from a series of bernouli proccesses

What I have are n different time series of bernouli processes of varying lengths, taking the values of 0 or 1. What I would like to do is to use Bayesian inference to calculate, for one of these ...
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### Conjugate Hyperpriors

I heard it was possible to have a Bayesian model with likelihood, prior and hyperprior that has a posterior of closed form, by choosing a conjugate prior and conjugate hyperprior. But I struggle to ...
1 vote
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### Beta-Binomial mixture vs Beta-Binomial multilevel model?

I first read about the Beta PDF in the context that it was conjugate to the Binomial distribution; a Beta prior with a Binomial likelihood returns a Beta posterior. So this sounds to me like a ...
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### Posterior distribution of $\theta x^{\theta - 1}$ with $Gamma(\alpha, \lambda)$ prior
Random variables $X_1, \ldots, X_n$ are i.i.d given $\vartheta = \theta$ and have the following pdf: \begin{equation} p(x|\theta)=\begin{cases} \theta x^{\theta - 1}, & \text{if $0<x<1$...