Questions tagged [jeffreys-prior]
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Why Is Jeffreys's Prior Used to Correct Biases?
I was reading this article on Logistic Regression for Rare Events.
Over here, a modification ("Firth's Correction") to the classical likelihood function has been proposed in which a penalty ...
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Quantiles of the posterior predictive distribution of a Gumbel random variable under the degenerate prior $\pi(\mu,\sigma) = \sigma^{-1}$
I need to find an automatic way to calculate with good precision the quantile of the posterior predictive distribution (ppd) of a random variable following a Gumbel law, under the degenerate prior $\...
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Jeffreys prior of a multivariate Gaussian
I have found two different expressions for the Jeffreys prior of a multivariate Gaussian. Eq. (3) in this article states that $$p(\mu,\Sigma) \propto \det(\Sigma)^{-(d+2)/2}$$
However in page 73 of ...
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Fisher information for the negative binomial distribution
I have the negative binomial distribution and want to find the fisher information: $I(\theta) = V[\ell'(\theta)]$
How do i calculate this?
I know that the derivative of the log-likelihood is: $\ell'(\...
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Derive posterior density function with Jeffrey's prior for theta
I need a guide on how to derive the posterior distribution for $\theta$ and checking whether it is proper. I have been given that the likelihood function is
$$L(\theta; x) = \theta \exp(−\theta x).$$
...
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Which form of Jeffrey's prior can be used for a three-parameter distribution?
Let X be a random variable which follows a distribution, say S with parameters a, b and c. Knowing that or Assuming that a, b and c are independent of one another, which one is reasonable to do?
a) Is ...
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Is there any strong argument about objective/non-informative improper prior?
Decades ago improper objective priors - e.g. $\pi(\sigma) \propto \sigma^{-1}, \sigma > 0,$ for a scale parameter - were considered problematic because some authors thought they were leading to the ...
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Understanding the Jeffreys-Zellner-Siow (JZS) prior in Bayesian t-tests
I am currently working on a lecture on Bayesian hypothesis testing, following the paper by Rouder et al, 2009. On Page 231 they present the formula for the relevant Bayes factor based on the Jeffreys-...
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Informative priors for standard deviation (or variance)
Suppose I want to perform Bayesian estimation of the mean $\mu$ and standard deviation $\sigma$ of a Gaussian distribution. Is there a standard way to specify an informative prior over $\sigma$, ...
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Finding the posterior under Jeffreys prior
Let $U \sim \text{Unif}(0,1)$, $\alpha$ the rate parameter of interest and $a>0$ a fixed known constant, define
$$ X = a U^{-1/\alpha}$$
Find the Jeffreys prior, the posterior for $\alpha$ and ...
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List of proper Jeffreys priors?
I know that Jeffreys priors are often improper. In fact, the only proper Jeffreys prior that I know is for the success probability in Bernoulli model (the prior arcsine).
I am curious to know if there ...
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In what noninformative priors turn out to be informative? [duplicate]
When searching about noninformative priors on internet, one can read here and there that those priors in fact turn out to be informative. However, I did not yet read a real argument about that.
So my ...
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Reference prior of normal distribution with unknow mean and variance
Problem:
Assume that $X|\theta \sim N(\theta, \sigma^2)$ for unknow $\theta$, and unknown $\sigma$.
a. Find the reference priors of $(\theta, \sigma)$, when $\sigma$ is of interest.
b. Find the ...
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Computing the Bayesian Estimator with Jeffreys prior for the Gamma distribution
Question:
Let $X_1, · · · , X_n$ be a random sample from $Gamma(1, θ)$. The population mean is $θ$. Assume that the Jeffreys prior is used.
Find the generalized Bayesian estimator of θ under the SEL (...
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How do I write the Jeffreys prior for error variances in stan? $p(\mu, \sigma_1^2, \dots, \sigma_C^2) \propto \Pi{ \sigma_i^{-2}}$ [closed]
I need to model the Jeffreys prior for error variances in a heteroscedastic ANOVA design in rstan.
That is to say, $\pi(\mu,\sigma_1^2,\dotsb,\sigma_C^2)\varpropto\Pi_{i=1}^{C}\sigma_i^{-2}$.
Is the ...
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Postetior from Jeffrey prior of Normal distribtion
Context
I am given a sample from normal distribution $v_i \sim N(\gamma \cdot u_i, \sigma^2)$, $i =1,..., n$.
I need to obtain the posterior distribution using Jeffreys prior for $\gamma$.
My solution
...
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Jeffreys' prior invariance under reparametrization
Let $X \sim Bin(n, \pi)$ with $\pi \in (0,1)$ and known $n$.
Derive Jeffreys' prior for $\pi$ in the following parametrizations:
a) Original parametrization $\pi$
b) Parametrization with $\phi = \text{...
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Posterior Distribution of a Normal Sample using Jeffreys Prior with a Known Parameter
Suppose I have a sample of $x_1, x_2, ... x_n$, where $X \sim N(\mu, \sigma^2)$, for some known $\sigma^2$, and that $\mu$ is defined only in $\mu \in [0, b]$, for some finite constant $b$.
It then ...
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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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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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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 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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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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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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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 $\pi(\mu|\sigma)$?
user238019
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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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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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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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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}
$$
approach?
I was trying to answer the ...
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What's the intuition for a Beta Distribution with alpha and / or beta less than 1?
I am curious for myself, but also trying to explain this to others.
The beta distribution is often used as a Bayesian conjugate prior for a binomial likelihood. It is often explained with the example ...
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Obtaining Jeffreys prior by taking the limit of a particular prior density on $(\mu, \Sigma)$
Text: Bayesian Data Analysis 3E by Gelman, section 3.6
Let $y | \mu, \Sigma \sim \text{MVN}(\mu, \Sigma),$ where
$\mu$ is a column vector of length $d$
$\Sigma$ is a $d \times d$ symmetric, ...
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Distribution for which the Jeffreys prior is Gaussian, log-normal, or exponential?
To be clear, I'm not looking for the Jeffreys prior on parameters of Gaussian, log-normal, or exponential distributions.
I am, instead, looking for a probability distribution, which has one or ...
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Proof question about Jeffreys' prior & normal distribution [closed]
Demonstrate that the Jeffreys' prior for the mean and variance parameters of normally distributed data $x=\{x_1,x_2,x_3,...,x_n\}$ is given by $p(\theta,\phi)\propto \phi^{-3/2}$.
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Jeffreys's prior for negative binomial regresion
For a negative biomial model, where $Y_i \sim \text{NegBin}(\mu_i, \kappa)$
$$\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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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 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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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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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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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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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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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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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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