Probit posterior score
= Probit posterior score
{title2=$g(\beta)$}
For <probit regression> with $s_i=2Y_i-1$ and prior $N(0,\sigma^2I)$, the <posterior score control variate> is
$$
g(\beta)=-\frac\beta{\sigma^2}+\sum_i s_ix_i\frac{\phi(s_ix_i^T\beta)}{\Phi(s_ix_i^T\beta)}.
$$
For $h(\beta)=\Phi(x_*^T\beta)$, its covariance with the score is $-x_*\mathbb E[\phi(x_*^T\beta)]$. It is nonzero whenever $x_*\ne0$, ensuring strict improvement by the optimal <control variate>.