Jeffreys prior

In Bayesian probability, the Jeffreys prior (called after Harold Jeffreys) is a
non-informative prior distribution proportional to the square rootof the Fisher information:

: $p(\; heta)\; propto\; sqrt\{I(\; heta\; |\; y)\}$

and is invariant under reparameterization of $heta$.

It's an important uninformative (objective) prior.

It allows us to describe our knowledge on $phi$, a transformation of $heta$ with an improper uniform distribution. This also implies the resulting likelihood function, $L(phi|X)$ should be asymptotically translated by changes in data. Due to asymptotical normality, this means only the first moment will vary when data is updated.

It can be derived as follows:

We need an injective transformation of $heta$ such that our prior under this transformation is uniform. It gives us "no information". We then use the following relation:

: $I(phi\; |\; y)\; =\; left(frac\{d\; heta\}\{dphi\}\; ight)^2I(\; heta\; |\; y)$

To conclude,

: $frac\{dphi\}\{d\; heta\}\; propto\; sqrt\{I(\; heta\; |\; y)\}$

: $phi\; propto\; int\_\; \{\}sqrt\{I(\; heta\; |\; y)\}\; d\; heta$

From a practical and mathematical standpoint, a valid reason to use this noninformative prior instead of others, like the ones obtained through a limit in conjugate families of distributions, is that it best represents the lack of knowledge when a certain parametric family is chosen, and it is linked with strong Bayesian statistics results.

In general, use of Jeffreys priors violates the likelihood principle; some statisticians therefore regard their use as unjustified.Fact|date=December 2007

References

*cite journal
last= Jeffreys | first=H. | authorlink=Harold Jeffreys
year = 1946
title = An Invariant Form for the Prior Probability in Estimation Problems
journal = Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
volume = 186
issue = 1007
pages = 453–461
url = http://links.jstor.org/sici?sici=0080-4630(19460924)186%3A1007%3C453%3AAIFFTP%3E2.0.CO%3B2-J
doi = 10.1098/rspa.1946.0056

*cite book
last= Jeffreys | first=H. | authorlink=Harold Jeffreys
year = 1939
title = Theory of Probability
publisher = Oxford University Press