# Difference between revisions of "Jeffreys prior"

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: <math>p\left(\vec\theta\right) \propto \sqrt{\det \mathcal{I}\left(\vec\theta\right)}.\,</math> | : <math>p\left(\vec\theta\right) \propto \sqrt{\det \mathcal{I}\left(\vec\theta\right)}.\,</math> | ||

It has the key feature that it is [[Parametrization#Parametrization invariance|invariant under reparameterization]] of the parameter vector <math>\vec\theta</math>. This makes it of special interest for use with ''scale parameters''.<ref>Jaynes, E. T. (1968) "Prior Probabilities", ''IEEE Trans. on Systems Science and Cybernetics'', '''SSC-4''', 227 [http://bayes.wustl.edu/etj/articles/prior.pdf pdf].</ref> | It has the key feature that it is [[Parametrization#Parametrization invariance|invariant under reparameterization]] of the parameter vector <math>\vec\theta</math>. This makes it of special interest for use with [[Scale_parameter|''scale parameters'']].<ref>Jaynes, E. T. (1968) "Prior Probabilities", ''IEEE Trans. on Systems Science and Cybernetics'', '''SSC-4''', 227 [http://bayes.wustl.edu/etj/articles/prior.pdf pdf].</ref> | ||

== Reparameterization == | == Reparameterization == | ||

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From a practical and mathematical standpoint, a valid reason to use this non-informative prior instead of others, like the ones obtained through a limit in conjugate families of distributions, is that it is not dependent upon the set of parameter variables that is chosen to describe parameter space. | From a practical and mathematical standpoint, a valid reason to use this non-informative prior instead of others, like the ones obtained through a limit in conjugate families of distributions, is that it is not dependent upon the set of parameter variables that is chosen to describe parameter space. | ||

Sometimes the Jeffreys prior cannot be [[Normalizing constant|normalized]], and thus | Sometimes the Jeffreys prior cannot be [[Normalizing constant|normalized]], and is thus an [[improper prior]]. For example, the Jeffreys prior for the distribution mean is uniform over the entire real line in the case of a [[Gaussian distribution]] of known variance. | ||

Use of the Jeffreys prior violates the strong version of the [[likelihood principle]], which is accepted by many, but by no means all, statisticians. When using the Jeffreys prior, inferences about <math>\vec\theta</math> depend not just on the probability of the observed data as a function of <math>\vec\theta</math>, but also on the universe of all possible experimental outcomes, as determined by the experimental design, because the Fisher information is computed from an expectation over the chosen universe. Accordingly, the Jeffreys prior, and hence the inferences made using it, may be different for two experiments involving the same <math>\vec\theta</math> parameter even when the likelihood functions for the two experiments are the same—a violation of the strong likelihood principle. | Use of the Jeffreys prior violates the strong version of the [[likelihood principle]], which is accepted by many, but by no means all, statisticians. When using the Jeffreys prior, inferences about <math>\vec\theta</math> depend not just on the probability of the observed data as a function of <math>\vec\theta</math>, but also on the universe of all possible experimental outcomes, as determined by the experimental design, because the Fisher information is computed from an expectation over the chosen universe. Accordingly, the Jeffreys prior, and hence the inferences made using it, may be different for two experiments involving the same <math>\vec\theta</math> parameter even when the likelihood functions for the two experiments are the same—a violation of the strong likelihood principle. | ||

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For the [[Gaussian distribution]] of the real value <math>x</math> | For the [[Gaussian distribution]] of the real value <math>x</math> | ||

: <math>f(x|\mu) = \frac{e^{-(x - \mu)^2 / 2\sigma^2}}{\sqrt{2 \pi \sigma^2}}</math> | : <math>f(x|\mu) = \frac{e^{-(x - \mu)^2 / 2\sigma^2}}{\sqrt{2 \pi \sigma^2}}</math> | ||

the Jeffreys prior for the mean <math>\mu</math> is | with <math>\sigma</math> fixed, the Jeffreys prior for the mean <math>\mu</math> is | ||

: <math>\begin{align} p(\mu) & \propto \sqrt{I(\mu)} | : <math>\begin{align} p(\mu) & \propto \sqrt{I(\mu)} | ||

= \sqrt{\operatorname{E}\!\left[ \left( \frac{d}{d\mu} \log f(x|\mu) \right)^2\right]} | = \sqrt{\operatorname{E}\!\left[ \left( \frac{d}{d\mu} \log f(x|\mu) \right)^2\right]} | ||

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For the [[Gaussian distribution]] of the real value <math>x</math> | For the [[Gaussian distribution]] of the real value <math>x</math> | ||

: <math>f(x|\sigma) = \frac{e^{-(x - \mu)^2 / 2 \sigma^2}}{\sqrt{2 \pi \sigma^2}},</math> | : <math>f(x|\sigma) = \frac{e^{-(x - \mu)^2 / 2 \sigma^2}}{\sqrt{2 \pi \sigma^2}},</math> | ||

the Jeffreys prior for the standard deviation σ > 0 is | with <math>\mu</math> fixed, the Jeffreys prior for the standard deviation σ > 0 is | ||

: <math>\begin{align}p(\sigma) & \propto \sqrt{I(\sigma)} | : <math>\begin{align}p(\sigma) & \propto \sqrt{I(\sigma)} | ||

= \sqrt{\operatorname{E}\!\left[ \left( \frac{d}{d\sigma} \log f(x|\sigma) \right)^2\right]} | = \sqrt{\operatorname{E}\!\left[ \left( \frac{d}{d\sigma} \log f(x|\sigma) \right)^2\right]} | ||

= \sqrt{\operatorname{E}\!\left[ \left( \frac{(x - \mu)^2-\sigma^2}{\sigma^3} \right)^2 \right]} \\ | = \sqrt{\operatorname{E}\!\left[ \left( \frac{(x - \mu)^2-\sigma^2}{\sigma^3} \right)^2 \right]} \\ | ||

& = \sqrt{\int_{-\infty}^{+\infty} f(x|\ | & = \sqrt{\int_{-\infty}^{+\infty} f(x|\sigma)\left(\frac{(x-\mu)^2-\sigma^2}{\sigma^3}\right)^2 dx} | ||

= \sqrt{\frac{2}{\sigma^2}} | = \sqrt{\frac{2}{\sigma^2}} | ||

\propto \frac{1}{\sigma}. | \propto \frac{1}{\sigma}. | ||

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the Jeffreys prior for the rate parameter λ ≥ 0 is | the Jeffreys prior for the rate parameter λ ≥ 0 is | ||

: <math>\begin{align}p(\lambda) &\propto \sqrt{I(\lambda)} | : <math>\begin{align}p(\lambda) &\propto \sqrt{I(\lambda)} | ||

= \sqrt{\operatorname{E}\!\left[ \left( \frac{d}{d\lambda} \log f( | = \sqrt{\operatorname{E}\!\left[ \left( \frac{d}{d\lambda} \log f(n|\lambda) \right)^2\right]} | ||

= \sqrt{\operatorname{E}\!\left[ \left( \frac{n-\lambda}{\lambda} \right)^2\right]} \\ | = \sqrt{\operatorname{E}\!\left[ \left( \frac{n-\lambda}{\lambda} \right)^2\right]} \\ | ||

& = \sqrt{\sum_{n=0}^{+\infty} f(n|\lambda) \left( \frac{n-\lambda}{\lambda} \right)^2} | & = \sqrt{\sum_{n=0}^{+\infty} f(n|\lambda) \left( \frac{n-\lambda}{\lambda} \right)^2} | ||

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===''N''-sided die with biased probabilities=== | ===''N''-sided die with biased probabilities=== | ||

Similarly, for a throw of an <math>N</math>-sided die with outcome probabilities <math>\vec{\gamma} = (\gamma_1, \ldots, \gamma_N)</math>, each non-negative and satisfying <math>\sum_{i=1}^N \gamma_i = 1</math>, the Jeffreys prior for <math>\vec{\gamma}</math> is the [[Dirichlet distribution]] with all (alpha) parameters set to | Similarly, for a throw of an <math>N</math>-sided die with outcome probabilities <math>\vec{\gamma} = (\gamma_1, \ldots, \gamma_N)</math>, each non-negative and satisfying <math>\sum_{i=1}^N \gamma_i = 1</math>, the Jeffreys prior for <math>\vec{\gamma}</math> is the [[Dirichlet distribution]] with all (alpha) parameters set to one half. In particular, if we write <math>\gamma_i = {\phi_i}^2</math> for each <math>i</math>, then the Jeffreys prior for <math>\vec{\phi}</math> is uniform on the (''N''–1)-dimensional [[unit sphere]] (''i.e.'', it is uniform on the surface of an ''N''-dimensional [[unit sphere|unit ball]]). | ||

==References== | ==References== | ||

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[[Category:Bayesian statistics]] | [[Category:Bayesian statistics]] | ||

## Latest revision as of 08:40, 21 May 2014

In Bayesian probability, the **Jeffreys prior**, named after Harold Jeffreys, is a non-informative (objective) prior distribution on parameter space that is proportional to the square root of the determinant of the Fisher information:

It has the key feature that it is invariant under reparameterization of the parameter vector . This makes it of special interest for use with *scale parameters*.^{[1]}

## Reparameterization

### One-parameter case

For an alternate parameterization we can derive

from

using the change of variables theorem and the definition of Fisher information:

### Multiple-parameter case

For an alternate parameterization we can derive

from

using the change of variables theorem, the definition of Fisher information, and that the product of determinants is the determinant of the matrix product:

## Attributes

From a practical and mathematical standpoint, a valid reason to use this non-informative prior instead of others, like the ones obtained through a limit in conjugate families of distributions, is that it is not dependent upon the set of parameter variables that is chosen to describe parameter space.

Sometimes the Jeffreys prior cannot be normalized, and is thus an improper prior. For example, the Jeffreys prior for the distribution mean is uniform over the entire real line in the case of a Gaussian distribution of known variance.

Use of the Jeffreys prior violates the strong version of the likelihood principle, which is accepted by many, but by no means all, statisticians. When using the Jeffreys prior, inferences about depend not just on the probability of the observed data as a function of , but also on the universe of all possible experimental outcomes, as determined by the experimental design, because the Fisher information is computed from an expectation over the chosen universe. Accordingly, the Jeffreys prior, and hence the inferences made using it, may be different for two experiments involving the same parameter even when the likelihood functions for the two experiments are the same—a violation of the strong likelihood principle.

## Minimum description length

In the minimum description length approach to statistics the goal is to describe data as compactly as possible where the length of a description is measured in bits of the code used. For a parametric family of distributions one compares a code with the best code based on one of the distributions in the parameterized family. The main result is that in exponential families, asymptotically for large sample size, the code based on the distribution that is a mixture of the elements in the exponential family with the Jeffreys prior is optimal. This result holds if one restricts the parameter set to a compact subset in the interior of the full parameter space. If the full parameter is used a modified version of the result should be used.

## Examples

The Jeffreys prior for a parameter (or a set of parameters) depends upon the statistical model.

### Gaussian distribution with mean parameter

For the Gaussian distribution of the real value

with fixed, the Jeffreys prior for the mean is

That is, the Jeffreys prior for does not depend upon ; it is the unnormalized uniform distribution on the real line — the distribution that is 1 (or some other fixed constant) for all points. This is an improper prior, and is, up to the choice of constant, the unique *translation*-invariant distribution on the reals (the Haar measure with respect to addition of reals), corresponding to the mean being a measure of *location* and translation-invariance corresponding to no information about location.

### Gaussian distribution with standard deviation parameter

For the Gaussian distribution of the real value

with fixed, the Jeffreys prior for the standard deviation σ > 0 is

Equivalently, the Jeffreys prior for log σ^{2} (or log σ) is the unnormalized uniform distribution on the real line, and thus this distribution is also known as the **Template:Visible anchor**. It is the unique (up to a multiple) prior (on the positive reals) that is *scale*-invariant (the Haar measure with respect to multiplication of positive reals), corresponding to the standard deviation being a measure of *scale* and scale-invariance corresponding to no information about scale. As with the uniform distribution on the reals, it is an improper prior.

### Poisson distribution with rate parameter

For the Poisson distribution of the non-negative integer ,

the Jeffreys prior for the rate parameter λ ≥ 0 is

Equivalently, the Jeffreys prior for is the unnormalized uniform distribution on the non-negative real line.

### Bernoulli trial

For a coin that is "heads" with probability γ ∈ [0,1] and is "tails" with probability 1 − γ, for a given (H,T) ∈ {(0,1), (1,0)} the probability is . The Jeffreys prior for the parameter is

This is the arcsine distribution and is a beta distribution with . Furthermore, if the Jeffreys prior for is uniform in the interval . Equivalently, is uniform on the whole circle .

*N*-sided die with biased probabilities

Similarly, for a throw of an -sided die with outcome probabilities , each non-negative and satisfying , the Jeffreys prior for is the Dirichlet distribution with all (alpha) parameters set to one half. In particular, if we write for each , then the Jeffreys prior for is uniform on the (*N*–1)-dimensional unit sphere (*i.e.*, it is uniform on the surface of an *N*-dimensional unit ball).

## References

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