Gaussian Probabilities and Expectation Propagation

TL;DR

This paper explores the use of Expectation Propagation (EP) for approximating Gaussian probabilities, demonstrating high accuracy in simple cases but revealing potential inaccuracies in more complex polyhedral regions, supported by empirical and theoretical analysis.

Contribution

It extends EP-based approximation methods to Gaussian probability calculations over polyhedral regions and analyzes their accuracy and limitations.

Findings

EP provides highly accurate approximations for rectangular regions.

EP's accuracy can be arbitrarily poor in polyhedral regions.

The paper offers theoretical insights into EP's behavior in complex integration regions.

Abstract

While Gaussian probability densities are omnipresent in applied mathematics, Gaussian cumulative probabilities are hard to calculate in any but the univariate case. We study the utility of Expectation Propagation (EP) as an approximate integration method for this problem. For rectangular integration regions, the approximation is highly accurate. We also extend the derivations to the more general case of polyhedral integration regions. However, we find that in this polyhedral case, EP's answer, though often accurate, can be almost arbitrarily wrong. We consider these unexpected results empirically and theoretically, both for the problem of Gaussian probabilities and for EP more generally. These results elucidate an interesting and non-obvious feature of EP not yet studied in detail.