Questions tagged [jeffreys-prior]

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Information Theoretical optimality of Jeffreys Prior?

Given a "conditional distribution" $f(x|\theta)$ and its corresponding Jeffreys prior $r(\theta)$, is there some information theoretic sense (in terms of quantities like mutual information) ...
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39 views

Jeffreys prior on invariant likelihoods

For a likelihood $p(y | \theta)$ and pdf $f(y)$: Suppose that a likelihood is location invariant i.e. $p(y | \theta) = f(y - \theta)$ Show that the Jeffreys prior is of the form $p(θ) ∝ 1$. I ...
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Prove that location/scale-invariant model is Jeffreys' prior

location model: $p(y|\theta)=f(y-\theta)$ how can i proof that $p(\theta) \propto 1$ scale model: $p(y|\theta) = \frac{1}{\theta} f(\frac{y}{\theta})$ how can i proof that $p(\theta) \propto \frac{...
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Jeffreys' prior for Bernoulli sampling

I am learning Bayesian Statistics and I don't understand the Jeffreys' prior for Bernoulli sampling below: If I understood well s is the number of observations when x=1 and f=n-s , where n is the ...
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87 views

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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43 views

Jeffreys prior vs. Flat prior on $(\beta,\log\sigma^2)$

I'm reading Bayesian Core, and the authors state that a Jeffreys prior $\pi(\beta,\sigma^2|X)\propto\frac{1}{\sigma^2}$ corresponds to a flat prior on $(\beta,\log\sigma^2)$. Why is this so?
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35 views

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 ...
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196 views

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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46 views

Are there situations where improper priors can be avoided via a prior on a subset of the real line and a transformation?

There are many situations where improper priors are "permissable" (Berger, 2009). In many cases, these improper priors are improper because they are "flat" on the real line. A well known example is ...
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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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806 views

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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Objective Bayesianism: Jeffreys priors vs reference priors vs principle of transformation groups

According to this answer, José Bernardo has produced an original theory of reference priors where he chooses the prior in order to maximise the information brought by the data by maximising the ...
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116 views

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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73 views

How to specify a new joint prior distribution restricted to (0,1) interval in WinBUGS?

I am doing some Bayesian analysis in OpenBUGS and I need to specify a joint prior distribution for parameters $\gamma_1$ and $\gamma_2$ (the Jeffreys prior related to my model), such that $$\pi(\...
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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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2k views

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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180 views

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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155 views

Distribution for which the Jeffreys prior is Gaussian, log-normal, or exponential?

To be clear, I'm not looking for the Jeffreys prior on parameters of Gaussian, log-normal, or exponential distributions. I am, instead, looking for a probability distribution, which has one or ...
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639 views

Proof question about Jeffreys' prior & normal distribution [closed]

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 $p(\theta,\phi)\propto \phi^{-3/2}$.
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263 views

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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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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1answer
112 views

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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416 views

Jeffreys' Prior for Log Odds

If you have a binomial likelihood, $y|n,p\sim\textsf{Bin}(n,p)$, the Jeffreys' prior for the proportion $p$ is $\textsf{Beta}(1/2,1/2)$. If we instead reparameterize the proportion as the log odds $\...
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47 views

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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7k views

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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1k views

When using Jeffrey's prior for Normal model, what is $p_J(\theta, \sigma^{2} | y_{1}, …, y_{n})$ supposed to be?

I'm reading A First Course in Bayesian Statistical Methods by P. Hoff where he is using Jeffrey's prior (J) and Unit information prior (U) for Normal model. For example we can derive Jeffrey's prior ...
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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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1answer
101 views

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\...
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1k views

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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857 views

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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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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Jeffreys prior for multiple parameters

In certain cases, the Jeffreys prior for a full multidimensional model is generaly considered as inadequate, this is for example the case in: $$ y_i=\mu + \varepsilon_i \, , $$ (where $\varepsilon \...
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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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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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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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171 views

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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83 views

Noninformative prior for variance

In Bishop's Pattern Recognition and Machine learning, he states that if you want a noninformative prior for scale,$$\int_A^Bp(\sigma)d\sigma = \int_{A/c}^{B/c}p(\sigma)d\sigma = \int_A^Bp\left(\frac{1}...
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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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200 views

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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1k views

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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76 views

Prior distributions letting a small sample “speak”

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 ...
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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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132 views

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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2k views

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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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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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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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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367 views

Using variance from a Jeffreys prior in a Prevalence meta-analysis

I am doing a meta-analysis on a group of studies, where the observation is a rate (no. of successes / no of trials). Some of the studies have small sample sizes (n<10) and/or did not observe any ...
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4k views

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 ...