Skip to main content

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
4 votes
1 answer
133 views

Conjugate Prior for Student T distribution with known degrees of freedom

Somebody asked a question about a conjugate prior distribution for Student-t distribution with unknown degrees of freedom. It was answered that there are no conjugate prior distribution in that case. ...
davynci's user avatar
  • 43
3 votes
0 answers
95 views

Form of conjugate prior on $(\mu,\sigma^2)$ for Normal $N(\mu,\sigma^2)$ distribution

I have seen many postings related to finding a conjugate prior for $(\mu,\sigma^2)$ for the normal distribution $N(\mu,\sigma^2)$. I am trying to derived an expression that has a rather simple ...
Oliver Díaz's user avatar
0 votes
0 answers
13 views

Posterior distribution for multivariate Gamma-Normal model

Let $\theta \in \mathbb{R}_{>0}^n$ be a random variable with prior distribution $p(\theta)$: \begin{equation} p(\theta) = \prod_{i=1}^n \text{Ga}(\alpha_i, \beta_i)(\theta_i), \end{equation} where $...
Mathieu le provost's user avatar
0 votes
0 answers
25 views

Conjugate or VB update for normal likelihood with shifted variance

Consider following model with normal likelihood at time $t$. \begin{align*} y_t | \mu_t, \sigma_t^2 \sim \mathcal{N}(\mu_t, \sigma_t^2 + P_t) \end{align*} we would like to find some priors $\mathcal{F}...
StatsyBanksy's user avatar
0 votes
0 answers
29 views

Conditionally conjugate prior in heteroskedastic model

I am researching a linear model where the noise is a function of the slope parameter as follows $$y_i = \beta_0 + \beta_1x_i + \beta_1\epsilon_i$$ $$\epsilon_i \sim N(0, \sigma^2 g)$$ where $g$ is ...
spencergw's user avatar
  • 141
5 votes
2 answers
413 views

How good is the Beta distribution as a conjugate for Binomial distribution?

I understand that the Beta Distribution is a 'natural conjugate' of the Binomial distribution, in sense that the Posterior Distribution is proportional to the multiplication of both. $$ Posterior(\...
Oscar Flores's user avatar
0 votes
0 answers
40 views

Posterior of Inverse Wishart distribution with a subset of data observed

Suppose: \begin{equation} x_1\in \mathbb{R}^{p_1}\\ x_2\in \mathbb{R}^{p_2} \end{equation} such that \begin{equation} x \sim \mathcal{N}( \begin{bmatrix} x_1\\ x_2 \end{bmatrix}; \begin{bmatrix} \...
Snowy Baboon's user avatar
2 votes
0 answers
46 views

Mean of normal follows a T distribution

Suppose: $x \sim \mathcal{N}(x; \mu, \Sigma) \;\;\;$ st. $\;\;\; \mu \sim T_{v}(\mu; k, M)$ Where $T$ is the $t$-distribution with v degrees of freedom, location $k$, and shape $M$. Then, is there a ...
Snowy Baboon's user avatar
1 vote
1 answer
60 views

Picking parameters for beta prior

I have some data that I believe come from a binomially distributed population. A beta prior seems like an appropriate choice, but I don't have any very strong prior beliefs. I could use a less ...
TerryStone's user avatar
2 votes
2 answers
248 views

What is the posterior probability for flipping a coin, assuming a beta distribution as conjugate prior

Suppose, I toss a fair coin n = 10 times and get 7 heads and 3 tails. The probability of fair coin is p = 0.5. Now, that the beta distribution is a conjugate prior of the binomial likelihood. I used ...
triangular_triffle's user avatar
2 votes
1 answer
21 views

Show that the multinomial distribution with $k$ categories and Dirichlet distribution are conjugate prior

Problem: Show that the following distributions are conjugate priors for the corresponding densities.. The multinomial distribution with $k$ categories and $$ p_{X|\theta_1 , \dots, \theta_k} (x_1, \...
Oskar's user avatar
  • 245
0 votes
0 answers
97 views

posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision

What is the posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision. Thus, what is the distribution of x given: \begin{equation} x \sim \mathcal{N}(x; \...
Snowy Baboon's user avatar
0 votes
0 answers
35 views

Conjugate prior for univariate normal with same mean and unknown sum of two variances

I have a Bayesian inference problem where the likelihood function is conditioned on two unknown variances. $$\log\mathcal{L}(d\mid \sigma_n,\sigma_s) = -\frac{1}{2} \log (\sigma_n^2 + \sigma_s^2) -\...
Riccardo Buscicchio's user avatar
0 votes
0 answers
94 views

Posterior Distribution using a Normal Likelihood and Laplace Prior

I have the working out below but is this correct. I just want the posterior distribution of when mu=0 given x. What I have tried is setting mu=0 after rewriting the first pdf with the summation ...
jjjcjjj893's user avatar
2 votes
0 answers
84 views

A hierarchical model with conjugate hyperprior

I have a modeling problem that I am trying to formulate in a Bayesian manner to do inference. Basically, I have a prior where the variance is unknown, and we want to treat it as uncertain (though with ...
smallStackBigFlow's user avatar
1 vote
0 answers
35 views

Bayesian conjugate updating when the likelihood can be approximated by a finite mixture of normals?

I'm facing a situation where I'd like to do Bayesian conjugate updating, but both the prior and the likelihood (a Student-t) can only be approximated by a finite mixture of normals. I know that a ...
Björn's user avatar
  • 32.8k
4 votes
1 answer
210 views

Is Inverse-Wishart a conjugate prior for Wishart likelihood?

Suppose I have a noisy observation $Z$ of a covariance matrix $F$, given a prior on $F: p(F)$, I would like to find the posterior of $p(F|Z)$, does the following specification forms conjugacy?: $$ F \...
K C's user avatar
  • 51
1 vote
0 answers
17 views

To derive Posteriors from Conjugate Priors, do we just multiply the terms in the PDFs with the parameters of interest?

Consider the beta-binomial model (beta prior, binomial likelihood). So we have$$ \begin{align} P(\theta)&\sim \text{Beta}(\theta|\hat a,\hat b) \propto \theta^{\hat a-1}(1-\theta)^{\hat b-1}\\ P(Y|...
user1176663's user avatar
1 vote
0 answers
153 views

Multivariate Normal Bayesian Updating with Conjugate Priors but Non-Standard Likelihood

I am trying to solve for the posterior of two parameters $\theta_1$ and $\theta_2$. I have priors $N(\mu_1, \sigma_1^2)$ and $N(\mu_2, \sigma_2^2)$ for $\theta_1$ and $\theta_2$ respectively, where $\...
bark's user avatar
  • 11
0 votes
0 answers
133 views

Bayesian Inference on a Normal Distribution. Unknown Variance, Known Mean. Scaled Inverse Chi squared as a Conjugate Prior

A common prior for the unknown variance of a normal distribution (with given mean $\mu$) is the scaled inverse $\chi-$squared distribution. It turns out this is a conjugate prior! A simple proof can ...
σκουλήκι's user avatar
2 votes
0 answers
273 views

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 ...
Björn's user avatar
  • 32.8k
3 votes
1 answer
729 views

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{\...
Hermi's user avatar
  • 727
0 votes
1 answer
90 views

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 ...
euphoric's user avatar
0 votes
0 answers
37 views

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 $\...
R S's user avatar
  • 537
0 votes
0 answers
282 views

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 ...
Leonidas's user avatar
  • 121
3 votes
1 answer
408 views

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 $...
fountain3's user avatar
0 votes
1 answer
2k views

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 ...
Dan W's user avatar
  • 183
0 votes
1 answer
57 views

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 ...
Valerio's user avatar
  • 37
4 votes
0 answers
931 views

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 ...
Homer Jay Simpson's user avatar
0 votes
0 answers
176 views

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 ...
Homer Jay Simpson's user avatar
0 votes
1 answer
85 views

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(\...
nbogs's user avatar
  • 101
1 vote
1 answer
190 views

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 ...
user1747134's user avatar
0 votes
1 answer
475 views

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 ...
user avatar
1 vote
1 answer
601 views

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 ...
Pseudodifferential Operators's user avatar
2 votes
0 answers
165 views

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 ...
user3269's user avatar
  • 5,182
0 votes
1 answer
217 views

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 ...
MoneyPrinting's user avatar
1 vote
0 answers
63 views

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\...
Elouan's user avatar
  • 11
6 votes
3 answers
1k views

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 ...
wd violet's user avatar
  • 777
2 votes
0 answers
37 views

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 ...
Turion's user avatar
  • 210
0 votes
0 answers
62 views

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 ...
user avatar
0 votes
1 answer
69 views

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 ...
SebastianP's user avatar
1 vote
2 answers
797 views

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 ...
Max Montana's user avatar
0 votes
0 answers
20 views

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 ...
Doosheck's user avatar
0 votes
0 answers
87 views

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 $\theta$...
RRMT's user avatar
  • 362
2 votes
0 answers
216 views

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 ...
Closed Limelike Curves's user avatar
0 votes
0 answers
140 views

References for the conjugate prior to the beta distribution? [duplicate]

The Wikipedia article about "Conjugate Prior" has a table containing information about Likelihood Distributions with their Conjugate Priors. In the "Continuous Likelihood" table, ...
Gilga's user avatar
  • 1
1 vote
1 answer
142 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 ...
Fernando Irarrázaval G's user avatar
0 votes
0 answers
46 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 ...
OLGJ's user avatar
  • 337
4 votes
2 answers
593 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 ...
Yair Daon's user avatar
  • 2,614
1 vote
1 answer
203 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'...
David's user avatar
  • 145

1
2 3 4 5
7