= Jeffreys prior for an additive variance component
{c}
{title2=$\pi(v)\propto\sqrt{\frac12\sum_s(v+r_s)^{-2}}$}
For independent normal observations with fixed means and variances $v+r_s$, where $v\ge0$ is unknown and all known $r_s>0$, the scalar <Fisher information> for $v$ is $I_{vv}=\frac12\sum_s(v+r_s)^{-2}$. 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.
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