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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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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85 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\...
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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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Interpretation of Gamma parameters in conjugate prior

So I'm modeling a problem using a Poisson process with rate $\theta$ and this $\theta$ follows a Gamma distribution with parameters $\alpha$ and $\beta$. Say $\theta$ captures the arrival of customers ...
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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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118 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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Bayesian Inference Intuition: Beta and Binomial vs Gamma and Poisson

When the data is assumed to be binomial distributed, and the prior probability is assumed to be a beta distribution, the posterior follows the distribution $Beta (\alpha - 1 + k , \beta - 1 + n- k $). ...
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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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Are there conjugate priors used in nonparametric density estimation methods?

I am still quite new to Bayesian inference and non-parametric methods. I was wondering if conjugate priors feature in non-parametrix density estimation. I understand that in non-parametric density ...
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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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83 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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Evaluating the integral using conjugate distributions

Hello I just want to verify that I am evaluating this integral correctly. When I implement it in code values seem to be incorrect. could be my implementation though. Thank you for the second pair of ...
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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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Is there a limiting distribution if you keep taking conjugate priors within the exponential family?

I'm assuming this is pretty commonly discussed but haven't found it yet. For example, if you start with a binary (0-1 binomial), conjugate prior is beta distribution. Beta distribution has some un-...
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Are $\alpha, \beta$ of Beta distribution positive integer inn Thompson Sampling

In wikipedia on beta distribution, they say that domain of hyperparameter $\alpha, \beta$ are positive real numbers. However, according to my reasons, the domain of $\alpha, \beta$ should be limited ...
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118 views

What family is the posterior predictive distribution in when the likelihood is a Bernoulli and the prior (and posterior) is Gaussian?

I have a problem where the hidden random variables I want to know about are continuous (i.e. my parameters can take any real number), and so I have modeled them as having a Gaussian p.d.f. (because ...
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Bayesian estimation difference between inferences is conjugate prior and non informative prior [duplicate]

What are the differences in the inferences using non-informative (specifically Jeffrys) priors as compared to the conjugate prior?
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Conjugate priors outside exponential family

The usual exception I have come across regarding non-existence of conjugate prior outside the exponential family is the uniform distribution on $(0,\theta)$ (i.e. $U(0,\theta)$) where $\theta$ has a ...
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Can a non-conjugate prior lead to a closed form or analytically tractable posterior? [duplicate]

I understand conjugate priors to the likelihood lead to a nice closed form posterior, and all members of the exponential family have conjugate priors. But is this if and only if? ie. Can a non-...
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What is the origin of the name “conjugate prior”?

I know what a conjugate prior is. But I'm confused by the name itself. Why is it called "conjugate"? A complex conjugate $z^\ast$ has a reciprocal relationship with $z$, i.e., ${z^\ast}^\ast ...
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Use inference on the whole data as a prior to regularize estimates in a subgroup

Suppose that the probability of a certain condition in the data $D$ is $\theta$ which can be estimated by updating a Beta prior as $Pr(\theta|s, N) = Beta(\alpha + s, \beta + N - s)$ with $s$ being ...
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How to use scipy stats gamma pdf to update the posterior distribution?

I'm trying to "get my bearings" performing bayesian analysis, specifically I'm exploring the Gamma-Poisson conjugate prior. The definition of the PDF is below If the prior takes the form of ...
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Beta-Binomial conjugate proof

Can someone explain this proof to me? I get stuck on the transition from the third line to the last line. Namely: Is the integral being evaluated or not? How does the entire expression reduce to a ...
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Direct relationship between prior and posterior parameter

I've been trying to learn JAGS and Bayesian modeling more generally and I'm running into something I can't quite explain. I've noticed that when fitting simple mean and variances to normally ...
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Bayesian priors and probability distributions

Book "Bayesian Statistics the Fun Way: Understanding Statistics and Probability with Star Wars, Lego, and Rubber Ducks", chapter 9 "Bayesian priors and working with probability ...
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Why is the inverse chi-squared distribution a natural prior and posterior for an unknown variance of a normal distribution?

Wikipedia says [the inverse-chi-squared distribution] arises in Bayesian inference, where it can be used as the prior and posterior distribution for an unknown variance of the normal distribution. ...
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The prior in MAP and Bayesian interference

We can use a Normal distribution as a prior when handling a Normal distribution as likelihood in Bayesian inference However if we want to do MAP given a Bernoulli as likelihood can we use Normal ...
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What is the conjugate prior for the hypoexponential distribution?

Can't find it anywhere. I know Gamma is the conjugate prior for the exponential distribution (one parameter) but for the sum of exponential distributions (the hypoexponential distribution), I can't ...
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Help understanding bayesian update for exponential distribution

I'm following these slides. I have some lamps, which I expect to die at a time $t$ where $p(t) = \lambda e ^{-\lambda t}$ for some $\lambda$. My prior for $\lambda$ is given by $p(\lambda) = \Gamma(\...
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Show that Inverse-gamma prior in Weibull distribution is conjugate

Weibull distribution: $fx(x) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}\text{exp}\left\{-\left(\frac{x}{\lambda}\right)^k\right\}$ Inverse gamma: $fx(x)=\frac{\beta^\alpha}{\Gamma(\alpha)...
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How to define different priors for $\mu$ parameters using a Normal-Inverse-Wishart prior?

Suppose, $y_i \sim N_2 (\mu, \Sigma)$ I want to define a Normal-Inverse-Wishart prior on $\mu$ and $\Sigma$. Such that, $(\mu, \Sigma) \sim NIW(\mu_0,\lambda,\Psi,v)$ My prior knowledge about $\mu ...
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Normal distribution with known mean and unknown variance (product of two variables)

Assume there is a data point $x$ sampled from a Normal distribution: $$\begin{align} x \sim \mathcal{N}(\mu,\frac{1}{yz}) \propto (yz)^{1/2} \exp [-\frac{1}{2} (x-\mu)^2yz] \end{align}$$ where $\mu$ ...
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Justification for use of non-conjugate priors?

Google searches gives no results to this question and there is the opposite question in this site, which makes me think this has an intuitive response I am missing. In most course notes and responses ...
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257 views

What form of conjugate prior best fits this likelihood distribution? [closed]

Joint likelihood of a two part model consisting of logistic regression and log-normal model:
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Bayesian prior that two parameters are identical/similar, but no information on their values?

Given two coins with respective biases $\mu_a$ and $\mu_b$, suppose that we have no information on their biases, but we believe that the two biases are identical or similar. Is there a standard/...
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Joint Posterior for Binomial Likelihood and Beta Priors

Suppose we have the likelihood for known $n$ $$\mathbf{x} \vert p,k \sim \mathrm{Binomial}(n, kp)$$ with a beta prior for $p$ with known parameters $a$ and $b$ $$p \sim \mathrm{Beta}(a, b)$$ and ...
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Gamma distribution and hyperparameters

The formula for mean and variance of a gamma distribution is given by a/b and a/b^2 (hyperparameters) respectively.Are they estimates of the posterior gamma distribution? Can prior, likelihood and ...
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Transformation of Beta distribution prior parameters for group testing analysis

As described here Optimization of pool size and number of tests for prevalence estimation via group testing I'm trying to estimate the prevalence of a lab test in a population (PCR for COVID-19) ...
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Posterior probability distribution from multinomial sample

I want to get the posterior from a multinomial sample and want to know if the following derivation is correct. Suppose when drawing (with replacement) a sample of $N$ balls from a urn with $K$ ...
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46 views

Posterior Distribution of Beta Prior

I have three values ($x_1=2$, $x_2=8$, and $x_3=4$). These are drawn from a Binomial distribution with parameter $k=12$ and unknown parameter $p$. The prior distribution of $p$ is a Beta distribution ...
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Are analytically tractable posterior distributions exclusively the result of a conjugate relationship in Bayesian hierarchical models?

I have been building a few of my own MCMC algorithms for hierarchical Bayesian models. If the posterior distribution of say $\alpha$ is analytically tractable, I sample $\alpha$ using an R function ...

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