For an alternative parameterization we can derive from using the change of variables theorem for transformations and the definition of Fisher information:
Multiple-parameter case
For an alternative parameterization we can derive from using the change of variables theorem for transformations, 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 the relative probability of a volume of the probability space 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 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 for all points. This is an improper prior, and is, up to the choice of constant, the unique translation-invariant distribution on the reals, corresponding to the mean being a measure of location and translation-invariance corresponding to no information about location.
For the Gaussian distribution of the real value with fixed, the Jeffreys prior for the standard deviation is Equivalently, the Jeffreys prior for is the unnormalized uniform distribution on the real line, and thus this distribution is also known as the . Similarly, the Jeffreys prior for is also uniform. It is the unique prior that is scale-invariant, 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.
For the Poisson distribution of the non-negative integer, the Jeffreys prior for the rate parameter 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 and is "tails" with probability, for a given 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 parameters set to one half. This amounts to using a pseudocount of one half for each possible outcome. Equivalently, if we write for each, then the Jeffreys prior for is uniform on the -dimensional unit sphere.
