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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Obtain Bayes estimator with conjugate prior

Consider n observations $ X_1, X_2,....X_n $ from $ Beta_1 ~ B(1,\theta ) $ distribution. Obtain Bayes estimator for $ \theta $ under quadratic loss function when conjugate prior is assumed for $\...
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Deriving Posterior with Nomalizing Flows

Typically, a Normal distribution is a conjugate prior for $\mu$ of a Normal distribution, we have a closed-form solution to update realize the Bayesian update. For example in Bayesian linear ...
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If the prior and likelihood not be conjugate, how to get conditional distribution to sample from using Gibbs sampling?

I know that when prior is conjugate with the posterior, by writing the loglikelihood and log prior and eliminate the non-independent terms for each parameter one can get the conditional distribution ...
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What if the prior not be conjugate with posterior in Bayesian learning?

I know that when the prior is conjugate with posterior then one can get an analytical representation for the posterior distribution, but what if these two are not to be conjugate? For example, I would ...
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Given a mean, what is the range of variance values that make for possible Beta distribution parameters

The beta distribution can have its parameter estimated via method of moments, which I will be doing. $$\hat\alpha = \bigg(\dfrac{\bar x (1-\bar x)}{var(X)} - 1\bigg)\bar{x}\\ \hat\beta = \bigg(\dfrac{\...
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How can the marginal distribution be derived from conjugate Gaussians?

In An Introduction to Empirical Bayes Data Analysis by George Casella (1985), it is given that \begin{align} x|\theta &\sim N(\theta,\sigma^2) \\ \theta &\sim N(\mu,\tau^2) \end{align} and ...
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How to compute the mean of two conjugate distributions in an analytic posterior distribution?

I do not know why in the following picture the mean of this posterior is $\mu_n= (X^TX+\Lambda_0)^{-1}(X^TX\hat{\beta}+\Lambda_0\mu_0)$
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Correlated belief update: Is this understanding of Bayesian posterior wrong

I am reading this paper Knowledge-Gradient Policy for Correlated Normal Beliefs for Rank and Selection Problem. The idea is as follows: We have $M$ distinct alternatives and samples from alternative $...
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Bayesian Weighted Least Squares Regression - Conjugate Prior with known correlation structure

I found this video (https://www.youtube.com/watch?v=LL3Dx79DIRw) which discusses a particular formulation of Weighted Least Squares Regression in a Baysian perspective. The model is: $$ y \sim Normal(...
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Likelihood and Prior density scales

I have a question about priors and likelihoods and their visualisation. A Bernoulli likelihood is $$\theta^{N_1}(1 - \theta)^{N_0}$$ where $N_1$ and $N_0$ are number of success and failures, ...
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Bayesian inference data from independent normal two different prior assumptions

Say we have $n$ data points $x_1, x_2, ..., x_n$, let's also assume that each data point comes from a normal distribution. For Bayesian inference, the first prior assumption is, for each data point $...
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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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Determining the exact posterior distribution

I am trying to find the exact posterior distribution from a prior that has the form $$ p(\theta) = [cos(4\pi\theta) + 1]^2$$ and the likelihood has the form $$ p(D|\theta) = \theta^n(1 - \theta)^{(N-n)...
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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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Tractable predictive distributions

I have a parametric model $p\left(x\mid\theta\right)$, the marginal distribution this induces is $$p\left(x\right)=\int p\left(x\mid\theta\right)p\left(\theta\right)d\theta$$ I understand the concept ...
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Exponential likelihood with scale parameter + Exponential prior with rate parameter

We have an exponential model with a scale parameter $\lambda$: $$ f(x|\lambda) = \frac{1}{\lambda}e^{-\frac{x}{\lambda}}1_{x>0} $$ We posit a prior on $\lambda$ with exponential distribution and ...
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HMM with emission probability being hidden state

I've been working on this problem for a while but cannot think of any solution. So here's the problem explained in the context of HMM. The hidden state is a probability that is updated given ...
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Recommended/Commonly Used Likelihoods for TF-IDF Observations in Mixture Models?

What are recommended or commonly used likelihoods for TF-IDF observations in mixture models? The below related questions ask about whether a Multinomial likelihood can be used (and if I understand ...
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Posterior predictive of normal normal-mean conjugacy

I want to compute: $$p(x | X) = \int p(x | \mu , \Sigma) p(\mu | X) = \int \mathcal{N}(x | \mu , \Sigma) \mathcal{N}(\mu | \mu_N , \Sigma_N ) d\mu$$ Actually, this is the posterior predictive of the ...
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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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Gamma family as conjugate prior of Inverse Gaussian with known $\mu$

I want to show that, when $\mu=\mu_0$, then gamma family $\Gamma(a,b)$ is a conjugate prior to inverse Gaussian with density $f(x,\mu,\lambda)=\sqrt{\frac{\lambda}{2\pi x^2}}exp[-\frac{\lambda(x-\mu)^...
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Bayesian Estimation of CDF

i'm getting pretty confused by the following problem, hope anyone can clarify my mind: Using a bayesian approach obtain a posteriori and interval estimations for $\mathbf{F}_{X}(x)$ using a Uniform(0,...
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proof for posterior predictive of normal-gamma conjugacy

Giving the following equations $$ \mu_n = \frac{\kappa_0 \mu_0 + n \overline{x}}{\kappa_0 + n}, \\ \kappa_n = \kappa_0 + n, \\ \alpha_n = \alpha_0 + n/2, \\ \beta_n = \beta_0 + \frac{1}{2} \sum\...
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161 views

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 ...
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What are the hyperparameters and base measure in the conjugate prior for the exponential family?

Setup Suppose we have an exponential family model $\{P_{\theta} : \theta \in \Theta\}$. Let the density function of a random variable $X$ and the prior on $\theta$ have following forms: $$ \begin{...
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Conjugate Prior Distribution with a Normally distributed marginal distribution?

The Normal-Inverse-Gamma distribution $(X,\mu,\sigma^2)\sim NIG(\mu_0, \nu, \alpha, \beta)$ is the conjugate prior for the Normal distribution. However, this would correspond to the marginal ...
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665 views

Non-informative prior for Exponential

I am working with a Bayesian model: $T \sim exp(\theta)$ for survival data, I have chosen a gamma distribution as a prior since its conjugate by an exponential distribution. I'd like to choose a $\...
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Does a conjugate prior always exists? [duplicate]

Are there distributions where no conjugate prior exists? Is there a necessary and/or sufficient condition which guarantees the existence of a conjugate prior? Edit: Why has this question been closed? ...
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Compound beta-binomial and beta distribution

I have a process that is modelled by a beta-binomial, parametrised by mean $\mu$ and correlation $\rho = 1/(\alpha+\beta+1)$ (as per dbetabinom in the R VGAM package). I know $\rho$, but the mean $\...
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Alternating between Hamiltonian Monte Carlo and sampling from conjugate posterior in large models

For Bayesian models with a large number of parameters and pieces that have conditionally closed form solutions for drawing samples, when is it worth it to use these closed form updates instead of a ...
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Normal Conjugate Prior, Known Mean and Unknown Variance?

For Normal distribution, with know mean and unknown variance. When $\tau = 1/\sigma^2$ ~ Gamma(). In such has posterior of $\tau$ has the following distribution: $p(\tau|\alpha, \beta, x) \sim G(\...
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Bayesian Regression Estimates

Hi I am new to Bayesian Regression, I wanted to understand why would the Bayesian regression give exactly the same results as the priors supplied? I tried running a bayesian model on 10% of the data ...
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137 views

Find the prior distribution for the natural parameter of an exponential family

Show that for the binomial likelihood $y$ ~$Bin(n, \theta)$, $p(\theta) \propto \theta^{-1} (1-\theta)^{-1}$ is the uniform prior distribution for the natural parameter of the exponential family. I am ...
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Random number generation for conjugate distribution of beta distribution

I try to generate random numbers from the conjugate distribution of beta distribution. It is as follows $$ p(α,β∣a,b,d)∝ \frac{e^{-a \alpha} e^{-b \beta}}{(\beta(\alpha,\beta))^d} \:\:\:\:,\:\:\: \...
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Help with the prior distribution

The question is as follows: Consider an SDOF mass-spring system. The value of the mass is known and is equal to 1 kg. The value of the spring stiffness is unknow and based on the experience and ...

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