  We consider the problem of estimating a probability of failure $\alpha$,
defined as the volume of the excursion set of a function $f:\mathbb{X}
\subseteq \mathbb{R}^{d} \to \mathbb{R}$ above a given threshold, under a given
probability measure on $\mathbb{X}$. In this article, we combine the popular
subset simulation algorithm (Au and Beck, Probab. Eng. Mech. 2001) and our
sequential Bayesian approach for the estimation of a probability of failure
(Bect, Ginsbourger, Li, Picheny and Vazquez, Stat. Comput. 2012). This makes it
possible to estimate $\alpha$ when the number of evaluations of $f$ is very
limited and $\alpha$ is very small. The resulting algorithm is called Bayesian
subset simulation (BSS). A key idea, as in the subset simulation algorithm, is
to estimate the probabilities of a sequence of excursion sets of $f$ above
intermediate thresholds, using a sequential Monte Carlo (SMC) approach. A
Gaussian process prior on $f$ is used to define the sequence of densities
targeted by the SMC algorithm, and drive the selection of evaluation points of
$f$ to estimate the intermediate probabilities. Adaptive procedures are
proposed to determine the intermediate thresholds and the number of evaluations
to be carried out at each stage of the algorithm. Numerical experiments
illustrate that BSS achieves significant savings in the number of function
evaluations with respect to other Monte Carlo approaches.
