= Reparameterization invariance of Jeffreys prior
{title2=$\sqrt{I_\eta(\eta)}=\sqrt{I_\theta(\theta)}\,|d\theta/d\eta|$}
For a regular scalar model, let $\eta=h(\theta)$ be a smooth invertible change of <statistical parameter> with nonzero <derivative>. The <chain rule> makes the <score function> in the new coordinate equal to the old <score function> times $d\theta/d\eta$. Therefore the expected <Fisher information> transforms as $I_\eta=I_\theta(d\theta/d\eta)^2$. Taking its square root gives exactly the absolute change-of-variable factor required for a <prior density>.
Consequently, computing the <Jeffreys prior> directly in either coordinate produces the same measure after transformation. Its density is not numerically constant across coordinates, nor must either density be uniform. When the <prior distribution> is proper, compatible normalization gives identical <prior probabilities> for corresponding sets; the resulting <posterior distributions> also transform consistently. For an <improper prior> the equality concerns measure kernels up to a constant, and <posterior propriety> still needs checking.
Back to article page
