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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
29 views

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 ...
user avatar
0 votes
0 answers
17 views

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 ...
user avatar
  • 81
0 votes
0 answers
38 views

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 ...
user avatar
0 votes
0 answers
18 views

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 ...
user avatar
3 votes
2 answers
131 views

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 ...
user avatar
  • 2,376
1 vote
1 answer
58 views

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'...
user avatar
  • 135
2 votes
2 answers
60 views

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 (...
user avatar
  • 135
0 votes
0 answers
29 views

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 ...
user avatar
0 votes
0 answers
22 views

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 ...
user avatar
0 votes
0 answers
20 views

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 ...
user avatar
0 votes
1 answer
75 views

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 ...
user avatar
  • 179
1 vote
0 answers
58 views

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 ...
user avatar
1 vote
0 answers
18 views

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 ...
user avatar
0 votes
0 answers
20 views

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 ...
user avatar
1 vote
0 answers
16 views

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 \...
user avatar
1 vote
2 answers
83 views

Exponential family and conjugate priors

Is a distribution that belongs to the exponential family necessarily conjugate prior?
user avatar
  • 113
0 votes
1 answer
39 views

Can a prior be conjugate and noninformative at the same time?

And if so, could somebody give me a concrete example?
user avatar
2 votes
1 answer
31 views

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 ...
user avatar
  • 749
0 votes
0 answers
27 views

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-...
user avatar
  • 749
0 votes
0 answers
23 views

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 ...
user avatar
0 votes
2 answers
50 views

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 ...
user avatar
  • 121
0 votes
0 answers
28 views

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 ...
user avatar
  • 31
0 votes
0 answers
28 views

[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 ...
user avatar
1 vote
1 answer
83 views

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 ...
user avatar
  • 1,700
0 votes
0 answers
50 views

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) ...
user avatar
0 votes
0 answers
48 views

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 ...
user avatar
  • 101
1 vote
1 answer
154 views

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 $0<x<1$...
user avatar
  • 13
0 votes
1 answer
216 views

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 ...
user avatar
0 votes
0 answers
75 views

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. $p(\mathbf{x}) \propto \sum_{i=1}^M w_iN(\mathbf{x}|\mu_i,\Sigma_i)$. ...
user avatar
  • 1
0 votes
1 answer
130 views

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 $\...
user avatar
  • 437
0 votes
0 answers
33 views

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 ...
user avatar
0 votes
2 answers
85 views

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 ...
user avatar
  • 143
2 votes
1 answer
148 views

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 ...
user avatar
  • 143
1 vote
0 answers
41 views

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{\...
user avatar
  • 31.3k
3 votes
1 answer
89 views

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 ...
user avatar
  • 3,006
0 votes
0 answers
29 views

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 $\mu_n= (X^TX+\Lambda_0)^{-1}(X^TX\hat{\beta}+\Lambda_0\mu_0)$
user avatar
  • 143
0 votes
1 answer
75 views

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 $...
user avatar
  • 1,282
0 votes
0 answers
43 views

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(...
user avatar
2 votes
2 answers
166 views

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, ...
user avatar
  • 117
0 votes
1 answer
59 views

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 $...
user avatar
0 votes
0 answers
33 views

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 ...
user avatar
0 votes
0 answers
28 views

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 ...
user avatar
  • 11
0 votes
0 answers
49 views

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)...
user avatar
0 votes
0 answers
30 views

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 ...
user avatar
  • 1
0 votes
0 answers
57 views

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 ...
user avatar
  • 109
5 votes
1 answer
100 views

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 ...
user avatar
9 votes
1 answer
378 views

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 ...
user avatar
  • 345
1 vote
1 answer
120 views

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)^...
user avatar
  • 551
1 vote
1 answer
187 views

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,...
user avatar
4 votes
1 answer
408 views

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\...
user avatar
  • 117

1
2 3 4 5 6