JeffreysPriors

Jeffreys priors are non-informative priors in Bayesian statistics named after Harold Jeffreys. They are designed to minimize subjective influence by using the data model itself, through the Fisher information. A key feature is invariance under reparameterization: inferences should not depend on how the parameter is expressed.

Mathematically, for a parameter vector θ, the Jeffreys prior is proportional to the square root of the determinant of the Fisher information matrix: π(θ) ∝ sqrt(det I(θ)). Here I(θ) is defined as the expected negative second derivative of the log-likelihood with respect to θ. In the univariate case, π(θ) ∝ sqrt(I(θ)).

Common examples illustrate the idea. For a normal model with known variance, the Jeffreys prior for the mean μ is constant (flat). For a scale parameter σ, the Jeffreys prior is proportional to 1/σ. For a Bernoulli parameter p, π(p) ∝ 1/sqrt(p(1−p)), i.e., a Beta(1/2, 1/2) prior. In a normal model with unknown mean and unknown variance, the joint Jeffreys prior is proportional to 1/σ^2.

Practical considerations include that Jeffreys priors can be improper, potentially yielding improper posteriors if the data are not informative enough. They may also behave poorly in complex or high-dimensional models and under certain parameterizations. Consequently, researchers sometimes use alternative objective priors, such as reference priors, or adopt weakly informative priors to balance invariance with statistical stability.