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Posterior Distribution using Jeffreys prior

I'm trying to show that if $X_1, \cdots, X_n \stackrel{iid}{\sim} N(\mu, \sigma^2)$ with unknown $\mu$, $\sigma$ and the prior $\pi(\mu, \sigma^2) \propto 1/\sigma^2$ then the posterior distribution ...
S10000's user avatar
  • 183
5 votes
2 answers
306 views

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 ...
stats_noob's user avatar
3 votes
1 answer
691 views

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 ...
Tendero's user avatar
  • 956
1 vote
1 answer
724 views

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'(\...
0xcc's user avatar
  • 105
2 votes
1 answer
316 views

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 $$L(\theta; x) = \theta \exp(−\theta x).$$ ...
Alex Forester's user avatar
1 vote
1 answer
118 views

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 ...
RRMT's user avatar
  • 362
1 vote
1 answer
107 views

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 ...
Celi's user avatar
  • 51
6 votes
1 answer
1k views

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-...
András Aszódi's user avatar
4 votes
2 answers
1k views

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 $\sigma$, ...
Betterthan Kwora's user avatar
0 votes
0 answers
52 views

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 ...
BelwarDissengulp's user avatar
2 votes
0 answers
251 views

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 ...
Celi's user avatar
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0 votes
0 answers
18 views

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 ...
Celi's user avatar
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1 vote
0 answers
441 views

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 ...
ForestGump's user avatar
2 votes
1 answer
480 views

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 (...
ForestGump's user avatar
0 votes
1 answer
299 views

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 ...
Dexter SherloConan's user avatar
1 vote
0 answers
295 views

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 ...
student's user avatar
  • 261
3 votes
1 answer
851 views

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{...
Mevve's user avatar
  • 165
0 votes
0 answers
643 views

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 ...
NicTam's user avatar
  • 83
0 votes
1 answer
137 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 ...
Brandon Barry's user avatar
1 vote
2 answers
3k views

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 ...
Schrodinger's user avatar
4 votes
1 answer
709 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, ...
Warhawk1987's user avatar
0 votes
0 answers
285 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?
statian's user avatar
  • 439
1 vote
1 answer
75 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 ...
Davey's user avatar
  • 171
2 votes
1 answer
877 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 ...
Scavenger23's user avatar
0 votes
1 answer
69 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 ...
Arjen Robben's user avatar
2 votes
0 answers
88 views

Conditional distribution with Jeffreys’ Prior [closed]

If $\pi(\mu,\sigma)$ corresponds to $N(\mu,\sigma^2)\times\mu^{-1/2}$, what is $\pi(\mu|\sigma)$?
user avatar
6 votes
1 answer
8k 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 ...
user208618's user avatar
4 votes
1 answer
212 views

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 ...
user76284's user avatar
  • 993
3 votes
1 answer
340 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 ...
AI2.0's user avatar
  • 133
1 vote
0 answers
214 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(\...
Guilherme Oliveira's user avatar
3 votes
0 answers
103 views

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 ...
Taylor's user avatar
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7 votes
4 answers
7k 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 ...
Mike Williamson's user avatar
4 votes
2 answers
540 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, ...
SOULed_Outt's user avatar
6 votes
2 answers
307 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 ...
Arthur's user avatar
  • 448
1 vote
1 answer
2k 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 $p(\theta,\phi)\propto \phi^{-3/2}$.
Stephen Wong's user avatar
2 votes
0 answers
575 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 ...
Björn's user avatar
  • 33.5k
11 votes
1 answer
5k views

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 ...
quantumflash's user avatar
2 votes
1 answer
304 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 ...
user1868607's user avatar
3 votes
2 answers
1k 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 $\...
aleshing's user avatar
  • 1,598
2 votes
0 answers
60 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/...
Alexey Zaytsev's user avatar
14 votes
1 answer
17k 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 $\phi = \theta^2$ ...
ben18785's user avatar
  • 738
3 votes
1 answer
4k 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 ...
Marko Lalovic's user avatar
11 votes
1 answer
5k views

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. ...
Babak's user avatar
  • 269
0 votes
1 answer
140 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\...
melatonin15's user avatar
5 votes
1 answer
3k 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 ...
ezhao15's user avatar
  • 153
2 votes
2 answers
2k 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 ...
user138644's user avatar
2 votes
0 answers
260 views

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{...
Count Zero's user avatar
  • 1,029
3 votes
0 answers
2k views

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 ...
a.powell's user avatar
  • 1,077
14 votes
1 answer
2k views

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 \...
beuhbbb's user avatar
  • 5,063
19 votes
2 answers
2k views

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 ...
user1398057's user avatar
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