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

Let $S={\sigma }^2+{\tau }^2$. For one posterior draw, the Gaussian quadratic-exponential moment is finite precisely when $\tau <\sigma$, and thenBecause the $m$ posterior draws are independent,For $\tau \ge \sigma$ the second moment, and hence the variance, is infinite. The Harmonic mean estimator of Bayesian model evidence is therefore unstable in the usual diffuse-prior regime: posterior sampling does not adequately control the reciprocal likelihood in the posterior tails.