Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 18H (Statistics)

Under $H_0$,Let $z_{1-\alpha /2}={\Phi }^{-1}(1-\alpha /2)$. The standard two-sided level-$\alpha$ test rejects exactly whenFor the observed value $\overline{X}=\overline{x}$, its two-sided Gaussian p-value is

Conditionally on $\mu$,so

Under the continuous half of the prior, write $\overline{X}=\mu +Z$ with independentThus $\overline{X}\sim N(0,{\tau }^2+1/n)$ there. Mixing this density with the point-null density gives the marginal likelihood for a Gaussian point-null mixture

Bayes' formula divides the point-null contribution to the mixture density by the full marginal density:Sincethe posterior probability of a Gaussian point null is

For $n=100$ and $\tau =1$,Both are continuous, so when $|\overline{x}|$ is sufficiently small,

For large $|\overline{x}|$, the supplied normal-tail approximation givesMeanwhile,Since $5000/101<50$, the posterior point-null probability decays more slowly. Therefore, for sufficiently large $|\overline{x}|$,This reversal is the Jeffreys-Lindley paradox for a Gaussian point null.