  We consider Bayesian optimization of objective functions of the form $\rho[
F(x, W) ]$, where $F$ is a black-box expensive-to-evaluate function and $\rho$
denotes either the VaR or CVaR risk measure, computed with respect to the
randomness induced by the environmental random variable $W$. Such problems
arise in decision making under uncertainty, such as in portfolio optimization
and robust systems design. We propose a family of novel Bayesian optimization
algorithms that exploit the structure of the objective function to
substantially improve sampling efficiency. Instead of modeling the objective
function directly as is typical in Bayesian optimization, these algorithms
model $F$ as a Gaussian process, and use the implied posterior on the objective
function to decide which points to evaluate. We demonstrate the effectiveness
of our approach in a variety of numerical experiments.
