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 a Bernoulli random variable with success probability $0<\theta <1$, the score function is $(X-\theta )/(\theta (1-\theta ))$. The expected value of its square is $I(\theta )=1/(\theta (1-\theta ))$, since the variance of $X$ is $\theta (1-\theta )$. The Jeffreys prior is therefore the proper beta distribution $\mathrm{Beta}(1/2,1/2)$, also called the arcsine distribution. The Euler beta function has $B(1/2,1/2)=\pi$, giving the stated normalized prior density.

After $s$ successes in $n$ independent Bernoulli trials, Beta-binomial conjugacy gives posterior distribution $\mathrm{Beta}(s+1/2,n-s+1/2)$ and posterior predictive probability $(s+1/2)/(n+1)$ of one further success. Although its prior density diverges at both boundaries, its integral is finite.

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.

A Poisson distribution with mean $E\lambda$ and positive known exposure has Fisher information $E/\lambda$, so its Jeffreys prior kernel is ${\lambda }^{-1/2}$. The proper shape-rate gamma distribution $\mathrm{Gamma}(1/2,\epsilon )$ has kernel ${\lambda }^{-1/2}{e}^{-\epsilon \lambda }$, approximating the improper kernel where $\epsilon \lambda$ is small. Its posterior is $\mathrm{Gamma}(y+1/2,E+\epsilon )$, converging to $\mathrm{Gamma}(y+1/2,E)$. There is no normalized Jeffreys prior to which the proper priors converge weakly; this is kernel approximation and posterior convergence.

For a differentiable scale family with finite positive Fisher information, the score is $-[1+u(\log f{)}^{\prime }(u)]/\sigma$. Its squared expectation is a constant times ${\sigma }^{-2}$, yielding the displayed Jeffreys prior. This is an improper prior on the entire positive half-line.

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.