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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.

For a regular one-parameter sampling distribution, the Fisher information and Jeffreys prior areUnder the usual differentiation and integrability conditions, $I(\theta )=-{\mathbb{E}}_{\theta }[{\partial }_{\theta }^2\log p_Y(Y\mid \theta )]$. This prior distribution transforms as a density under smooth one-to-one reparameterizations, so the rule is coordinate invariant. Its integral need not be finite; posterior propriety must still be established if it is an improper prior.

For a binomial distribution, the score function isIts squared expected value is $I(\theta )=n/[\theta (1-\theta )]$, using the binomial distribution variance $n\theta (1-\theta )$. Hence the Jeffreys prior isthe Beta distribution $\mathrm{Beta}(1/2,1/2)$. The factor $\sqrt{n}$ is independent of $\theta$ and disappears on normalization.

Independent Jeffreys priors and the two independent binomial distribution likelihood factors give, by Beta-binomial conjugacy,The Bayesian posteriors remain independent because each observation factor involves only its own probability. Their posterior means are respectively $0.7$ and $0.3$. Notice that $p_T$ is the probability of saying milk first when tea was first, rather than the probability of a correct tea identification.

The Jeffreys prior is proportional to $[\theta (1-\theta ){]}^{-1/2}$, hence is $\mathrm{Beta}(1/2,1/2)$. Its posterior is $\mathrm{Beta}(X+1/2,n-X+1/2)$, givingThe interior case is used only when $n\ge 2$. Generally neither rule equals $X/n$: the mean agrees only at $X=n/2$; the mode agrees at the endpoints and, in the interior, at $X=n/2$. For the small experiments $n=1,2$, all possible observations happen to be among these cases, so the whole mode rule then coincides with the maximum-likelihood estimator.