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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Dirichlet Process posterior with partially observed data

Suppose I dipose of a set of independant observed couples $(x_1,y_1),...,(x_N, y_N)$ from a joint distribution $P(x,y)$. Furthermore, I suppose that the random distribution $P$ as a Dirichlet prior $P\...
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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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How should I deduce the conjugate prior and corresponding posterior for a geometric distribution

The given pmf is for a geometric distribution and is $f(x_i|\theta) = (1-\theta)^{x_i - 1}\theta; ~x_i = 1, 2 ,\cdots, $ and the 1-parameter exponential family I have obtained is; $$f(x|\theta) = \exp ...
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Prior on a dirichlet distribution [duplicate]

I would like to know if there is a "conjugate" prior that we can place on the Dirichlet distribution parameters. Thanks in advance,
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Posterior distribution when the domain of the likelihood depends on the parameter

I am trying to calculate a posterior density given distribution and a prior. And I am a bit confused about how I should act as the domain of the distribution depends on the parameter. I am talking ...
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Sequential Bayesian updating of mean and variance of normal distribution

I am trying to write some code to learn the parameters of a normal distribution. I am new to this, and I have patched together the equations from various sources, which may be part of the problem. In ...
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Posterior density problem

I am preparing for an exam and I have stumbled upon this exercise, which I am not certain of. a) You are given i.i.d. data $x_1, \dots, x_n$ from a continuous distribution with density $\frac{\alpha ...
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Identifying the value of parameters of the prior distribution. Arbitrarily?

Referring to this Question, let's not use Jeffrey's prior for $\theta$ but use $Gamma(\alpha,\beta)$ as the conjugate prior for $\theta$. Under quadratic loss function, the bayes estimator for $\theta$...
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Conjugate Prior for Multivariate Normal Variances and Correlations

Is there a way to separately specify conjugate priors for the variance and correlations of a multivariate normal? The inverse Wishart is conjugate if you want to specify the covariance, but covariance ...
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mean-field variational inference in Nonconjugate Models

I'm trying to conduct mean-field Variational Inference (VI) with Nonconjugate model. I found Variational Inference in ...
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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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The PDF of the Data Given (Marginal Likelihood) the Likelihood and the Prior of a Normal Distribution with Prior on the Mean

Given a model where $ x_i | \mu \sim \mathcal{N} ( \mu, \sigma^2 ) $ where $ \mu \sim \mathcal{N} ( \mu_0, \sigma_0^2 ) $, is there a closed form formula for the PDF of $ x_i $? Namely, what's $ p (...
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Bayesian A/B testing with normal conjugate model for huge (non-normal) sample sizes

I'm running an A/B test with 100k+ users per group. It consists of a lot of different metrics, some continuous, some counts... they are highly skewed, with a long tail. I'm mostly interested in the ...
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Derivation of posterior distribution under Dirichlet prior distribution:

suppose that $\mathbf{y}=(y_1, y_2, \cdots, y_n)$ is a vector of $n$ observed sample points drawn from a mixture of $g$ components, and $\mathbf{z}=(z_1, z_2, \cdots, z_n)$ is a vector of latent ...
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Sequential Bayesian updating with binary data (in a case where a beta-bernoulli setup seems inappropriate)

An unknown parameter $\theta$ is randomly drawn at time $t=0$ according to prior p.d.f. $\mu_0(\cdot)$ that has support $[L,R]\subseteq\mathbb{R}$. At each time $t\in\{1,2,...\}$ an agent makes an ...
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Moments of the natural statistics of the normal gamma

I am trying to find the Moments of the natural statistics of the normal-gamma distribution. $$(X,T) - NormalGamma(\mu, \lambda,\alpha,\beta)$$ I found on its Wikipedia page that the moments of the ...
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Modeling "Pay as Much as You Want" with a Bayesian Model

I have data of sales of a certain product which is sold "Pay as Much as You Want". The daily data is in the form of number of sales per day and the total revenue per day: Day Sales Revenue ...
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A Proper Conjugate Model for A/B Test for Revenue per Click (RPC)

What would be a proper Conjugate Posterior model for Earning / Revenue per Click in A/B test? The data is the total number of visitors and the total revenue per day per variant (A and B). What are the ...
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Help understanding gamma posterior of exponential likelihood

The posterior of $\text{Exp}(x;\lambda)$ with prior $\text{Gamma}(\lambda;\alpha, \beta)$ is $\text{Gamma}(\lambda|\alpha+n, \beta + n\bar x)$ where $n$ is the number of observations and $\bar x$ is ...
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Conjugacy for right censored data in survival analysis

In survival analysis is it possible to have conjugate priors for the likelihood of right-censored data? More precisely, the likelihood of right-censored data is of the form: $$P(X|\theta) = \Pi _{i \...
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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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Understanding conjugate priors

For a Normal likelihood, the conjugate prior is: Normal: if both mean and variance are known, or if mean is unknown and variance is known Inverse Gamma: if known mean and unknown variance Normal-...
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Choosing informative Gibbs priors for Bayesian updating

I'm trying to create some kind of iterative Bayesian algorithm, which continuously updates as more data is gathered. The aim is to iteratively update the coefficients based on the second dataset using ...
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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: $$p(x│\theta)=\theta\exp(-\theta x)$$ Note that the ...
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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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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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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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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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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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