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 applies.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 219 / 4 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-219/4/a/i/solution