Marginal likelihood for a Gaussian point-null mixture

For $\overline{X}\mid \mu \sim N(\mu ,1/n)$ and an equal mixture of $\mu =0$ and $\mu \sim N(0,{\tau }^2)$,

For the equal point-null mixture,

With a fixed diffuse alternative prior, a frequentist p-value and the posterior probability of a point null can order evidence differently. For $n=100$ and $\tau =1$, the p-value exceeds the null posterior near zero, while sufficiently far in the tails the null posterior exceeds the p-value because it has a slightly slower Gaussian decay.