Adaptive nested rare event simulation algorithms

Nested simulation algorithms are used in several scientific investigations such as climate, statistical mechanics, and financial and actuarial risk management. Recently, these methods have also been used in the context of Bayesian computations and are referred to as Nested Sampling. In several of these problems, the inner level computation typically involves simulating events with very small probability, leading to rare event importance sampling methods. The quality of the resulting estimates depend on the allocation of computational resources between inner and outer level simulations. We introduce a novel adaptive rare event simulation algorithm that allocates the computational resources by taking in to account marginal changes in the rare event probabilities. We establish the consistency and efficiency of our algorithm and theoretically and numerically compare our results with the non-adaptive methods. We illustrate the proposed methods with several examples.