## probit posterior mean

In a recent arXiv report, Yuzo Maruyma shows that the posterior expectation of a probit parameter has an almost closed form (under a flat prior), namely

$\mathbb{E}[\beta|X,y] = (X^TX)^{-1} X^T\{2\text{diag}(y)-I_n\}\omega(X,y)$

where ω involves the integration of two quadratic forms over the n-dimensional unit sphere. Although this does not help directly with the MCMC derivation of the full posterior, this is an interesting lemma which shows a closed proximity with the standard least square estimate in linear regression.