  This paper develops a multifidelity method that enables estimation of failure
probabilities for expensive-to-evaluate models via information fusion and
importance sampling. The presented general fusion method combines multiple
probability estimators with the goal of variance reduction. We use low-fidelity
models to derive biasing densities for importance sampling and then fuse the
importance sampling estimators such that the fused multifidelity estimator is
unbiased and has mean-squared error lower than or equal to that of any of the
importance sampling estimators alone. By fusing all available estimators, the
method circumvents the challenging problem of selecting the best biasing
density and using only that density for sampling. A rigorous analysis shows
that the fused estimator is optimal in the sense that it has minimal variance
amongst all possible combinations of the estimators. The asymptotic behavior of
the proposed method is demonstrated on a convection-diffusion-reaction partial
differential equation model for which $10^5$ samples can be afforded. To
illustrate the proposed method at scale, we consider a model of a free plane
jet and quantify how uncertainties at the flow inlet propagate to a quantity of
interest related to turbulent mixing. Compared to an importance sampling
estimator that uses the high-fidelity model alone, our multifidelity estimator
reduces the required CPU time by 65\% while achieving a similar coefficient of
variation.
