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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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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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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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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Do Bayesian estimators under square error loss have an invariance property?

I feel like this is something we went over in class but it's not coming to me for some reason. I need to find the Bayesian estimator for $\tau(\theta)=e^{-\theta}$ under square error loss. I already ...
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Use a Bayesian approach to construct a 95% confidence/credible interval for the mean and variance using conjugate priors

I am looking for information on how to solve this problem above in R. It does not have to be solved using any specific dataset but I would appreciate an example. I am unsure of conjugate priors. ...
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Why use MCMC sampling when using conjugate priors?

I've been getting to grips with some Bayesian modelling, but one thing is confusing the heck out of me when I look at tutorials and worked-through problems online. I'm looking at a problem with a ...
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Conjugate priors calculation [duplicate]

I am following the Bayesian Methods for Machine Learning course on Coursera. Unfortunately, it glosses over many details, and I am struggling to understand how to check if a distribution is a ...
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What is the intuitive meaning of these statements in the context Bayesian prior and posterior?

I am now familiar with the Bayesian thinking process of using a prior and then getting the posterior once we observe data using the prior. I read the following statements which I am trying to get my ...
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Specific step in the proof of conjugate prior for normal distribution with unknown mean and variance

I'm struggling to follow a specific step in the proof that $$ \tau \sim \text{Gamma}(\alpha, \beta), \quad \mu | \tau \sim \mathcal{N}(\nu, \frac{1}{k\tau}) $$ is a conjugate prior distribution for ...
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Deriving the binomial-beta conjugate model

I think I am correctly deriving the binomial-beta conjugate model, but my answer differs slightly from what's on Wikipedia's page on conjugacy. My solution Assume that $$ X_t \sim \text{Binomial}(m,...
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Bayesian factor analysis: Help with posterior derivation

I am trying to derive the Bayesian factor analysis model described on page 10 of this paper. In brief, consider the model (simplified from the paper) $y_i \sim N(M \boldsymbol{f}_i, S), \quad i=1,\...
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Estimate covariance given noisy data and the mean

Basically, I'm inferring the parameters of a Gaussian, $\mu$ and $\Sigma$ given observed data $y_i$ that have uncertainties $\sigma^2_i$ associated with them. 1D example: Prior I intend to use the ...
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Finding prior conjugate for reparametrized model

Let $X_i$ be iid Bernoulli$(\pi)$ for $i=1,...,n$. My task is to find the prior conjugate for $\theta$, where $\theta$ is the natural parameter of the sampling model. The sampling model can be ...
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71 views

Poisson-Gamma conjunction - calculating posterior [duplicate]

How to calculate posterior distribution step-by-step while given: some observed numbers of customers from the last days that number of clients is distributed by Poisson($\lambda$) ($\lambda$ is not ...
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What distributions are conjugate to themselves, besides the normal?

I know the normal distribution is conjugate to itself; are there others? Is there some sort of intuition behind why a given distribution would be conjugate to itself?
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MLE vs Binomial - Beta conjugate pairs

I'm trying to understand how to use the conjugate relationship between the Binomial and Beta distributions to update the parameters of the Beta distribution. Specifically, I am imagining a series of ...
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Bayesian A/B test analysis for non-binary actions

When analyzing experiments with binary action (for example, will a user convert or not), then Beta-Binomial distribution is apt for that. For example, to model Click Through Rate (CTR): ...
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Conjuage priors for linear combination

My knwoledge of statistics in general and Bayesian statistics in particular is limited. With that in mind, I would sincerely appreciate if somebody could help me with the following problem that I have ...
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A Bernoulli mixture model with a Dirichlet prior on the parameters

Let's $x_1,...,x_N$ be a set of observation coming from the following generative process: $$ \boldsymbol{\theta} \sim \text{Dirichlet}(\boldsymbol{\alpha})\qquad\boldsymbol{\theta},\boldsymbol\alpha\...
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Bayesian Linear Regression, trouble with posterior. Variance equal identity

I am trying to solve the following problem. If $y | \beta \sim N(X \beta, I_n)$ and $\beta \sim N(0, g^{-1}(X^t X)^{-1})$ for $g>0$. Find $ \pi(\beta|y)$ and show that $E(\beta|y)$ is a function ...
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Literature on Bayesian stuff with Normal Distribution? [closed]

I am writing something on Bayesian Analysis involving the normal distribution. I know that the conjugate prior is the so-called normalized Gamma inverse distribution, I know the update rule for the ...
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Gibbs sampling: Conjugate prior for a two component known-unknown variance?

If I have a normally distributed variable $\mathcal{N}(\mu,\frac{1}{\tau})$ with fixed $\mu$ then the conjugate prior for an unknown $\tau$ is then $\mathcal{Ga}(\frac{n}{2}+\alpha, \beta + \sum_i \...
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Posterior (conjugate) prior of two parameter Gamma likelihood

This question is related to a previous question on this site. Assume some data is generated from Gamma distribution $p(x\mid\alpha,\beta) \sim \operatorname{Gamma} (\alpha,\beta)$, and both parameters ...
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Posterior distribution for Weibull scale parameter with censored data

I am modeling a survival problem using a Weibull distribution with known shape parameter $k$ and unknown scale parameter $\theta$. The PDF and CDF are given by, \begin{align} f(t|k,\theta)=\frac{k}{\...
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what is the step by step procedure of updating a posterior probability with new data coming in

I have a question about how to update posterior probability sequentially when new data comes in sequentially by say $x_1$, then $x_2$, then $x_3$,.... I understand this form when the first data $x_1$ ...
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Closed form for Finite Gaussian Mixture Model when weights are known and prior variance can be 0

Suppose I have a normal likelihood $x|\theta \sim N(\theta, \sigma^2_{known})$ where the variance is known and a mixture prior $\theta \sim p * N(\mu_1, \sigma^2_1) + (1-p) * N(\mu_2, \sigma^2_2)$, ...
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What is the relation between “conjugate priors” and the approximate inference?

I know that "conjugate prior" is to help us calculate the the denominator of the Bayes formula(to make the calculations easier). And I just learnt to approximate the inference by mean field ...
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If $f(x|\theta)$ is conjugate to $p(\theta)$ then is $f(x|r\theta)$ conjugate to $p(\theta)$?

If exponential family $f(x|\theta)$ is conjugate to $p(\theta)$ then is $f(x|r\theta)$ for $r>0$ conjugate to $p(\theta)$? If not, what can we do about it in terms of sampling to make use of ...
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Is there a conjugate prior for normal location family N(x|u,1) such that the mean is always positive?

The conjugate prior to normal location family is usually a normal distribution. However, I want to constrain the mean to be positive. Is there a conjugate prior to the normal location family $x\sim N(...
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Shape of parameters marginal posterior in hierarchical Bayes model

Consider a generic hierarchical Bayes model with data $y_i\sim p(y|\theta_i)$, dependent of parameters $\theta_i\sim p(\theta|\phi)$ and hyperparameters $\phi\sim p(\phi)$. Furthermore, assume that $\...
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Tuning the “strength” of updates to a posterior distribution for conjugates

I'm asking this question as a sanity check- I am not a statistician or research scientist, and just am doing a gut check on a model I am building. I want to quantify uncertainty of a specific metric ...
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Metropolis-within-Gibbs for parametric inference of a regressive model

I have a regressive model of this form \begin{equation} y=f(\theta)+\varepsilon \end{equation} to describe observations $y$, with noise $\varepsilon$ and a parametric function $f$ with parameters $\...
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Determine hyper-prior for gaussian distribution from existing data [closed]

Not sure how to determine hyper-prior for prior distributions, specifically using historical data. First what I am doing: I want to estimate parameters for a normal likelihood function using Bayesian ...

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