Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 18H / b / Solution

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