= Solution
Using <Bayes theorem> and the fact that the prior is proper,
$$
\mathbb E_{\theta\mid y}\widehat I
=\int\frac1{p(y\mid\theta)}
\frac{p(y\mid\theta)p(\theta)}Z\,d\theta
=\frac1Z\int p(\theta)\,d\theta
=\frac1Z.
$$
Thus $\widehat I$ is an <unbiased estimator> of $Z^{-1}$, and the <Harmonic mean estimator of Bayesian model evidence> is $\widehat Z=1/\widehat I$. The reciprocal is not itself generally unbiased, though it is consistent when the <strong law of large numbers> applies.
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