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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Inconsistency between normal-wishart marginal and marginal covariance computed using tower rule

Murphy's result, section 8.3 on normal conjugacy states (substituting $\mathbf{y}$ for $\mu$) that if: $$\mathbf{y}|\Lambda \sim \mathcal N(\mathbf{0}, (\kappa\Lambda)^{-1}) $$ $$ \Lambda \sim \...
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Integral of normal likelihood and multivariate normal prior

I'm updating cluster assignments in the context of a non-parametric Bayesian mixture model. When computing the probability of starting a new cluster, in the absence of cluster parameters (and using a ...
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Posterior predictive distribution for Bernoulli (and categorical)

I'm trying to confirm something I've tried to figure out about the posterior predictive distribution for Bernoulli vs. Binomial (and categorical vs. multinomial) random variables after a Bayesian ...
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Bayesian estimation of iid sample from Uniform$[0,\theta]$ and a Pareto$(\alpha,\beta)$ prior for $\theta$

I am working on Bayesian estimation: suppose that $X_1,\dots, X_n$ is an iid sample from Uniform$[0,\theta]$. Assume a Pareto prior for $\theta\sim Pareto(\alpha,\beta)$, i.e. $$ f(\theta)=\frac{\...
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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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Is updating gamma conjugate distributions always increasing?

In Bayesian inference, the gamma distribution is the conjugate prior to many likelihood distributions: Poisson, exponential, normal (with known mean), etc. The update rule seems to always add to $\...
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Posterior distribution of a $\text{Gamma}(\alpha,\beta)$ random variable given a Gamma prior for $\beta$

Let $Y$ be a $\text{Gamma}(\alpha,\beta)$ random variable with known shape parameter $\alpha$ and unknown scale parameter $\beta$. Suppose we assign a $\Gamma(\alpha_0,\beta_0)$ prior to $\beta$. I am ...
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What is the conjugate prior for the Von Mises distribution's precision

Does the Von Mises distribution have a conjugate prior for its precision/variance? Update: The concentration parameter $\kappa$ (Kappa) seems to control the variance of the Von Mises distribution. If $...
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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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Should the updated posterior for a Poisson distribution be discretized if based on the Gamma distribution as the prior?

I know that the Gamma distribution is the conjugate prior of the Poisson distribution, such that given $\alpha$ and $\beta$ that describe the prior distribution, the posterior distribution is $Gamma(\...
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Does the beta negative binomial (BNB) distribution have a conjugate prior?

BNB distribution is constructed using negative binomial and beta distributions, which are both exponential family, so my guess would be yes, there shoudl exist a conjugate prior in theory. But what is ...
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Use the gamma prior to obtain the gamma posterior values

I have the following information for the ages of individuals: Sample size = 5. Data: $$ x_i = (10, 12, 15, 16, 14) $$ The population mean previously accurately estimated is 12. Prior information ...
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Find a conjugate prior for the Weibull distribution under reparametrization

Consider the Weibull sampling model for $X_1,\ldots,X_n$ iid, where $$p(x|\lambda,k)=k\lambda^kx^{k-1}e^{-\lambda^kx^k}$$ for $x>0$. Assume $k$ is known and $\lambda$ is unknown. First, if I adopt ...
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the reason for using InverseGamma and LogNormal as prior for covariance matrix or variance

In the Bayesian analysis, sometimes we can see that InverseGamma and LogNormal distributions are used as prior for variance or covariance matrix respectively. What are the logic or explanations of ...
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Proportional to Gamma means the posterior is gamma

I'm reading through these lecture notes on posteriors and conjugate priors. https://web.stanford.edu/class/stats200/Lecture20.pdf In particular, it asserts that: "This is proportional to the PDF ...
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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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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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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 ...
sam's user avatar
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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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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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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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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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