Jeffreys prior

The Jeffreys prior is a default prior distribution in Bayesian statistics named after Harold Jeffreys. It is constructed to be invariant under reparameterization — a property that makes it the natural "uninformative" prior when one wants the conclusions of inference to depend on the data, not on how the parameter was labeled. Mathematically, the Jeffreys prior is proportional to the square root of the determinant of the Fisher information matrix: p(θ) ∝ √det I(θ).\n\nThis construction reveals that the Jeffreys prior is not merely a conventional choice but a geometric one: it is the volume element of the statistical manifold equipped with the Fisher-Rao metric. What looks like a subjective Bayesian choice is, from the geometric perspective, as objective as measuring the surface area of a sphere. The prior is not an expression of belief but a measure of the intrinsic size of the parameter space.\n\nThe Jeffreys prior has limitations: it can be improper (not normalizable) for unbounded parameters, and it may perform poorly for multiparameter problems where the cross-parameter correlations in the Fisher information matrix produce counterintuitive shapes. In such cases, reference priors and other objective Bayesian constructions attempt to preserve invariance while addressing these pathologies.\n\n\n\n