# 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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### 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 ...
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### [Bayesian][Conjugate Priors] How to update gamma prior distribution using a sample

The true data is believed to come from a Poisson distribution and I want to use a Gamma to model it. I have my prior, a gamma distribution. I am then shown a sample of data that I am to use to update ...
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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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### Is There a Conjugate Prior for a Multivariate Hypergeometric Likelihood?

I am working on a problem using a multivariate hypergeometric likelihood. The multivariate hypergeometric distribution does not belong to the exponential family of distributions, so (to my knowledge) ...
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### What's the difference between using negative inverse gamma vs. inverse gamma as the conjugate prior distribution in bayesian analysis?

My current understanding is that inverse gamma is used as the conjugate prior distribution when the likelihood function is a normal distribution with known mean and unknown variance. What's the effect ...
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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 $0<x<1$...
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### What is the posterior distribution of θ? Is the Gamma a conjugate prior for an exponential likelihood?

A manufacturer is interested in the time to failure of his batteries. Suppose the time to failure of the batteries has an exponential distribution: p(x│θ)=θe^(-θx) Note that the mean of this ...
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### Conjugate prior bayesian inference on multivariate GMM

I am trying to understand how the posterior looks like when running Bayesian inference on a multivariate Gaussian-mixture model. $p(\mathbf{x}) \propto \sum_{i=1}^M w_iN(\mathbf{x}|\mu_i,\Sigma_i)$. ...
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### Conjugate prior for the gaussian distribution

Let a set of random observations $\{X_i\}$, such that $X_i \sim \mathcal{N}(\mu, \sigma^2)$. Suppose that the mean $\mu$ and the variance $\sigma^2$ both are unknown. What is the conjugate prior for ...
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### Technical term for natural statistics comined with log-partition

One way to find the posterior of an exponential family distribution with a conjugate prior is to use the natural reparametrization of the likelihood and prior and combining the sufficient statistics ...
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### Beta-binomial vs updating a prior beta distribution

Bear with me, as I've just recently been learning about conjugate priors, prior and posterior distributions, and such material. My understanding of the beta-binomial distribution is that it basically ...
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### Normal Conjugate Normal Inverse-gamma Updating

I am attempting to code a multi-arm bandit where there are multiple variants that can be served to customers with the objective of learning the best one based on an outcome modeled with a normal ...
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### Closed form posterior for a mixtures of two univariate Gaussians

Giving a univariate Gaussian mixture model $$\pi_1N(x|\mu_1,\sigma_1)+(1-\pi_1)N(x|\mu_2,\sigma_2),$$ are there any priors for $\pi_1$, $\mu_1$, $\sigma_1$, $\mu_2$, $\sigma_2$ which gives a closed ...
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### Are there any conjugate likelihood distributions for a Categorical Prior?

I have the following generative process: \begin{align} z &\sim Cat(\pi)\\ o | z &\sim p(o|z) \end{align} I'd like to infer a posterior over $z$ i.e. $p(z|o)$. Thankfully, I have complete ...