Simulating from, and computing probabilities under, the truncated multivariate normal distribution (the normal law restricted to a rectangle / linear inequalities) is a recurrent and hard problem in high dimensions. Botev introduces a minimax exponential tilting method that provides both accurate estimates of the (possibly tiny) truncation probability and exact iid draws from the truncated distribution. The proposal builds on Genz's separation-of-variables importance-sampling construction, but adds an exponential tilt whose parameters minimize the worst-case variance of the estimator; the resulting scheme achieves high acceptance rates in over 100 dimensions and controlled relative error even for rare-event probabilities.
"[We present a] tilting method for exact iid simulation from the truncated multivariate normal distribution. The new methodology provides both a … estimator [of the probability] and … [samples] … The importance sampling proposal builds on the celebrated Genz construction, but the addition of the minimax tilting ensures that the [worst-case variance is controlled]."
The definitive modern answer to a problem the GHK simulator popularized but never fully solved: how to compute high-dimensional Gaussian rectangle probabilities — and sample the corresponding truncated normal — accurately in the tails. The minimax framing is the elegant part: instead of hoping a proposal happens to have low variance for the integral at hand, it explicitly optimizes against the worst case, and log-concavity makes that optimization cheap. That it gives exact iid draws (not a Markov chain) at roughly one-Gibbs-cycle cost is what made it the default for truncated-MVN sampling in the TruncatedNormal software. On the wiki it is the successor on the GHK page and a drop-in upgrade wherever correlated limited-dependent-variable models (multinomial probit, multivariate probit) need Gaussian orthant probabilities.