 Jeffreys-Lindley paradox for a Gaussian point null

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.