Closed-form marginal for a moment (non-local) slab prior on a z-score

For z ~ N(theta, 1) with the moment prior pi(theta) = (theta^2/tau^2) N(theta; 0, tau^2), you don't need numerical convolution. The marginal is m(z) = N(z; 0, 1+tau^2) * (s2 + mu^2)/tau^2, with s2 = tau^2/(1+tau^2) and mu = s2*z, i.e. the normal-slab marginal times E[theta^2 | z]/tau^2 under the normal slab. It integrates to 1, and it makes spike-plus-slab selection models (truncate at |z| > 1.96, divide by P(selected)) cheap to fit by MLE for either slab. Get P(selected) for the moment slab by 1-D quadrature of m over |z| > c.