= Solution
Conditionally on $\mu$,
$$
\overline X\mid\mu\sim N\left(\mu,\frac1n\right),
$$
so
$$
\boxed{
f(\overline x\mid\mu)
=\sqrt{\frac n{2\pi}}
\exp\left[-\frac n2(\overline x-\mu)^2\right]}.
$$
Under the continuous half of the prior, write $\overline X=\mu+Z$ with independent
$$
\mu\sim N(0,\tau^2),
\qquad
Z\sim N(0,1/n).
$$
Thus $\overline X\sim N(0,\tau^2+1/n)$ there. Mixing this density with the point-null density gives the <marginal likelihood for a Gaussian point-null mixture>
$$
\boxed{
\begin{aligned}
m(\overline x)={}&\frac12\sqrt{\frac n{2\pi}}
e^{-n\overline x^2/2}\\
&+\frac1{2\sqrt{2\pi(\tau^2+1/n)}}
\exp\left[-\frac{\overline x^2}{2(\tau^2+1/n)}\right].
\end{aligned}}
$$
Back to article page
