Past exam of the statistics course of the University of Oxford / 2017 / 3 / c

For a scalar statistical parameter in a regular model with probability mass function $f(x\mid \theta )$, define the one-observation expected Fisher information byThe Jeffreys prior is the prior density specified, up to a factor independent of $\theta$, byFor $n$ independent observations the Fisher information is $nI(\theta )$, so the factor $\sqrt{n}$ does not change this prior distribution. If $\int \sqrt{I(\theta )}d\theta$ is finite, divide by this integral to normalize; otherwise the rule defines an improper prior.

The Jeffreys prior is invariant under smooth one-to-one reparameterization. This means that applying the rule directly in a new parameter $\eta =h(\theta )$ gives the same prior distribution as first constructing the rule for $\theta$ and then transforming its prior density by the change-of-variable formula.

More explicitly, for a differentiable invertible change of parameter with nonzero derivative,Thus ${\pi }_{J,\eta }(\eta )d\eta$ and ${\pi }_{J,\theta }(\theta )d\theta$ represent the same measure, up to the normalization constant. Invariance concerns the measure: the numerical values of the prior densities need not be equal in the two coordinates. The same change of variables then makes the posterior distributions agree. This is the reparameterization invariance of Jeffreys prior.

For one Bernoulli random variable,Its variance is $\theta (1-\theta )$, so the expected Fisher information and Jeffreys prior areThe normalizer is $B(1/2,1/2)=\pi$. ThereforeThis Jeffreys prior for a Bernoulli parameter is proper despite its integrable endpoint singularities. It is the arcsine distribution. It replaces the specified $a,b>1$ prior distribution in part (a) when the Jeffreys prior rule is chosen; it is not one of those restricted priors.