Jeffreys prior (${\pi }_J(\theta )\propto \sqrt{\det I(\theta )}$)

The Jeffreys prior uses the square root of the determinant of the Fisher information matrix as a parameter-density kernel. It is invariant under smooth one-to-one reparameterization, since the information and density Jacobians transform compatibly. It may be an improper prior; invariance does not guarantee posterior propriety. With nuisance parameters, a scalar conditional Jeffreys prior and the joint Jeffreys prior need not coincide.

For independent normal observations with fixed means and variances $v+r_s$, where $v\ge 0$ is unknown and all known $r_s>0$, the scalar Fisher information for $v$ is $I_{vv}=\frac{1}{2}{\sum }_s(v+r_s{)}^{-2}$. Thus its scalar Jeffreys prior is finite at zero and behaves like $1/v$ at infinity. Unlike a log-flat prior $1/v$, it avoids a divergent integral at the zero-variance boundary. After flat-prior elimination of a Gaussian common mean, the integrated likelihood is bounded by a constant times $v^{-(N-1)/2}$, so this prior yields a proper posterior for $N>1$ and a proper prior on any remaining mean-shape parameters. In the homoscedastic case it reduces to $1/(v+r)$, which is log-flat for the total variance rather than for the latent variance alone.