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# Questions tagged [jeffreys-prior]

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3 votes
0 answers
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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 ...
• 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 ...
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 ...
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1 vote
1 answer
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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 ...
• 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 ...
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 ...
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, ...
• 569
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?
• 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 ...
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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 ...
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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 ...
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 $\pi(\mu|\sigma)$?
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 ...
• 151
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 ...
• 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 ...
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1 vote
0 answers
214 views

• 281
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 ...
• 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 ...
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{...
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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 ...
• 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 \...
• 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 ...
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