Strong log-concavity in probit regression

We show that strong log-concavity emerges in probit regression likelihoods without ridge penalization (i.e. Gaussian priors), unlike for the logistic case. Specifically, we provide: (a) a characterization of strong log-concavity for fixed designs, similar to that for the existence of the maximum likelihood estimator (MLE) and (b) an analysis for Gaussian design, dependent on the proportionality $d\mathrm{/}n=r\in [0,1)$ between the sample size $n$ and the number of covariates $d$. In the latter case we show that, with high probability, provided $r$ is small enough, the resulting condition number is finite and, in the asymptotic regime $n,d\to \mathrm{\infty }$, independent of $r$.