# Questions tagged [jeffreys-prior]

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### Example for a prior, that unlike Jeffreys, leads to a posterior that is not invariant

I am reposting an "answer" to a question that I had given some two weeks ago here: Why is the Jeffreys prior useful? It really was a question (and I did not have the right to post comments at the time,...
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### What is the relation behind Jeffreys Priors and a variance stabilizing transformation?

I was reading about the Jeffreys prior on wikipedia: Jeffreys Prior and saw that after each example, it describes how a variance-stabilizing transformation turns the Jeffreys prior into a uniform ...
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### Jeffreys Prior for normal distribution with unknown mean and variance

I am reading up on prior distributions and I calculated Jeffreys prior for a sample of normally distributed random variables with unknown mean and unknown variance. According to my calculations, the ...
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### Do statisticians use the Jeffreys' prior in actual applied work?

When I learned about the Jeffreys' prior in my graduate statistical inference class my professors made it sound sort of like it was interesting mostly for historical reasons rather than because anyone ...
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### Jeffreys prior for binomial likelihood

If I use a Jeffreys prior for a binomial probability parameter $\theta$ then this implies using a $\theta \sim beta(1/2,1/2)$ distribution. If I transform to a new frame of reference $\phi = \theta^2$...
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### Jeffreys' prior for Beta distribution

If my likelihood has the form of a beta distribution, and I want to use Jeffreys' prior for its parameters, what is form of the prior? For some distributions its pretty straight forward to calculate. ...
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### Why is uniform prior on log(x) equal to 1/x prior on x?

I'm trying to understand Jeffreys prior. One application is for 'scale' variables like the standard deviation $\sigma$ (or its square, the variance $\sigma^2$) of Gaussian distributions. It is often ...
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### Parametrisation invariance/covariance of the Jeffreys prior

I've been trying to understand what exactly is meant by parametrisation invariance of the Jeffreys prior. Already I've read here that invariance is technically not the best term to use, and that it'...
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### How to construct “reference priors”?

I have been reading about noninformative priors. Two of the most popular priors of this kind seem to be the Jeffreys prior and the reference prior. The Jeffreys prior has a clear construction, being ...
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### Define own noninformative prior in stan

In the simple case of normally distributed data with unknown mean and variance, Jeffrey's prior is given by $$p(\mu, \sigma^2)=\frac{1}{\sigma^2}.$$ How can I define such a prior in the Stan language,...
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### How can a uniform prior make the posterior mean different from the MLE?

I read the following in Machine Learning: A Probabilistic Perspective: How can a uniform prior move the posterior mean? Isn't a uniform distribution supposed to not bias the result? Are there any ...
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### Posterior distribution for Gamma scale parameter under the Jeffreys prior

What is the posterior distribution for parameter $b$ with $X \sim Gamma(a,b)$, under the Jeffreys prior? We can assume that $a$ is known. The Jeffreys prior is the square of the Fisher information ...
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### Why is Gamma(0,0) equivalent to the Jeffreys prior

I'm trying to use some code that includes Gamma priors for Poisson (rate) and Exponential (rate) distributions. I want to make the priors noninformative. I read that using a Gamma(0,0) is equivalent ...
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### Understanding definition of informative and uninformative prior distribution

When using the "non-informative" prior $\pi(\mu,\sigma)\propto\frac{1}{\sigma^2}$ where $\pi(\mu)\propto1$ and $\pi(\sigma^2)\propto\frac{1}{\sigma^2}$ Where is the no information for the ...
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### What's the intuition for a Beta Distribution with alpha and / or beta less than 1?

I am curious for myself, but also trying to explain this to others. The beta distribution is often used as a Bayesian conjugate prior for a binomial likelihood. It is often explained with the example ...
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### Use of the Jeffreys prior in multidimensional models

Suppose a model, $$x_{i} \sim N(\theta_{i}, \phi), \text{ for } i=1,\ldots,n$$ Furthermore, suppose the variance parameter, $\phi$, is some known constant. The multidimensional Jeffreys prior is ...
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### What is a good example of a non-informative prior for the uniform distribution?

I recently noticed that for non-informative priors, people usually use something like a uniform prior, which works for many different distributions. However, assuming that your likelihood is nothing ...
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### Understanding the Proof for why Jeffreys' prior is invariant

I was reviewing the section of Andrew Gelman's "Bayesian Data Analysis" on uninformative priors, and came across this explanation for why Jeffreys' prior is invariant to parameterization. My question ...
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### What reparametrization of vector parameters makes the Jeffreys prior correspond to the uniform prior?

What reparametrization of vector of parameters $\theta$ makes the Jeffreys prior $$\sqrt{\det I(\theta)}$$ correspond to the uniform prior? A change of parametrization from $\theta$ to $\eta$ changes ...
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### What is the limit of this expression?

If $\det(\Lambda_0) \to 0$, what does $$\exp\left(-\frac{1}{2}\text{trace}\left(\Lambda_0 \Sigma^{-1}\right)\right)\det\left(\Lambda_0\right)^{-1/2}$$ approach? I was trying to answer the ...
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### How to choose a importance density for Jeffreys prior?

I want to draw Bayesian inference via importance sampling and I do not come up with a good idea of an importance density for $$p(\sigma)\sim\frac{1}{\sigma}.$$ Is there a way to sample from this ...
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### Compute $\pi(H_0|x)$ with Jeffreys prior for a family $N(\theta,1)$

Given a random sample $x = (x_1,\ldots,x_n)$ taken from a family $\{N(x|\theta,1):\theta \in \mathbb{R}\}$. And consider the hypothesis test: $H_0: \theta = 0$ vs $H_1: \theta \in \mathbb{R}$ (this ...
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### Jeffreys' prior on variance

Jeffreys' prior on variance (var.), although uninformative, is not flat, but it is equivalent to assuming that the logarithm of the variance is uniformly distributed on the real line. So: A) how I ...
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### Calculating Jeffreys Prior for geometric distribution

This question is already answered here, but I would like to know why it is worked out the way it is My lecture notes state the following: I am also given the following problem : Now, what I ...
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### Obtaining Jeffreys prior by taking the limit of a particular prior density on $(\mu, \Sigma)$

Text: Bayesian Data Analysis 3E by Gelman, section 3.6 Let $y | \mu, \Sigma \sim \text{MVN}(\mu, \Sigma),$ where $\mu$ is a column vector of length $d$ $\Sigma$ is a $d \times d$ symmetric, ...
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### Questions on issues with using Frequentist and Bayesian approach for the same test

One quick stats question, if I use Binomial Cumulative Distribution Function to get a sample size n for desired confidence level and tolerable error. Then we pick a sample of sample size n and find k ...
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### Reasoning regarding non-informative priors

I'm not sure whether this counts as a question. However, I'd be happy to receive feedback for the validity of my reasoning. Recently, I read a bit about Jeffreys' prior and the "problem" with using ...
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### Conditional distribution with Jeffreys’ Prior [closed]

If $\pi(\mu,\sigma)$ corresponds to $N(\mu,\sigma^2)\times\mu^{-1/2}$, what is $\pi(\mu|\sigma)$?
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### Optimal Shrinkage with g-prior: estimate alpha to minimize MSE

In one of the slides from my class related to bayesian linear regression, I have the following scenario. Under g-prior, the shrinkage estimator induced by the prior is \hat{\beta_{\alpha}} = \alpha\...
I’ve got a general question. Let k be a parameter which must be estimated. It lies within the interval $[a, b]$, $a$ and $b$ being finite real numbers. Let us further assume we dispose of a series ...