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