  This paper deals with the problem of estimating the volume of the excursion
set of a function $f:\mathbb{R}^d \to \mathbb{R}$ above a given threshold,
under a probability measure on $\mathbb{R}^d$ that is assumed to be known. In
the industrial world, this corresponds to the problem of estimating a
probability of failure of a system. When only an expensive-to-simulate model of
the system is available, the budget for simulations is usually severely limited
and therefore classical Monte Carlo methods ought to be avoided. One of the
main contributions of this article is to derive SUR (stepwise uncertainty
reduction) strategies from a Bayesian-theoretic formulation of the problem of
estimating a probability of failure. These sequential strategies use a Gaussian
process model of $f$ and aim at performing evaluations of $f$ as efficiently as
possible to infer the value of the probability of failure. We compare these
strategies to other strategies also based on a Gaussian process model for
estimating a probability of failure.
