# Questions tagged [jeffreys-prior]

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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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### 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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### 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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### 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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### 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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### 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 vs. Empirical Bayesian analysis

I have a small data set, provided at the very end, where I have computed Jeffreys Prior to being a Beta(.5,.5) distribution. I then use this Jeffreys prior to report a 95% posterior credible set, ...
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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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### Jeffreys's prior for negative binomial regresion

For a negative biomial model, where $Y_i \sim \text{NegBin}(\mu_i, \kappa)$ $$\mu_i:=\log EY_i = \mathbf{x_i} \mathbf{\beta} + \log t_i,$$ is the form of Jeffreys's prior known/published in some way ...
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### Equivalance of reference prior and Jeffreys prior for $d = 1$

There are a number of sources that mention that the reference prior is equivalent to Jeffreys prior for $d = 1$ (see e.g. https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/lectures/...
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### Understanding my posterior with an uninformative prior with a poisson likelihood. Am I thinking about this correctly?

I have a problem to which I am trying to apply a Bayesian model. My data is generated as follows \begin{align} N_i \mid \mu &\sim \text{Poisson}(\mu) \\ Y_i \mid N_i, \theta_i &\sim \text{...
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### Sample Size Formula for Wilson Score, Clopper Pearson, and Jeffrey's

I am interested in finding the sample size formulas for proportions using the Wilson Score, Clopper Pearson, and Jeffrey's methods to compare with the Wald method. Also if anyone has code to replicate ...
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### Is Independent jeffreys prior different from independent reference prior?

I have a model involving two scalar parameters $\theta_1$ and $\theta_2$ and derived the Jeffreys prior for $\theta_1$ and $\theta_2$ independently (so for, e.g. $\pi(\theta_1)$, setting in the ...
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### Example of a uniform prior not being objective

The key feature of a truly objective prior is that it is invariant under change of variables. I understand this concept, however, I'm having a hard time finding a simple 1D or 2D example of when you ...
Demonstrate that the Jeffreys' prior for the mean and variance parameters of normally distributed data $x=\{x_1,x_2,x_3,...,x_n\}$ is given by $p(\theta,\phi)\propto \phi^{-3/2}$.