  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.
