Rare event simulation and splitting for discontinuous random variables

Abstract

Abstract. Multilevel Splitting methods, also called Sequential Monte-Carlo or Subset Simulation, are widely used methods for estimating extreme probabilities of the form P [S(U) > q] where S is a deterministic real-valued function and U can be a random finite-or infinite-dimensional vector. Very often, X := S(U) is supposed to be a continuous random variable and a lot of theoretical results on the statistical behaviour of the estimator are now derived with this hypothesis. However, as soon as some threshold effect appears in S and/or U is discrete or mixed discrete/continuous this assumption does not hold any more and the estimator is not consistent.In this paper, we study the impact of discontinuities in the cdf of X and present three unbiased corrected estimators to handle them. These estimators do not require to know in advance if X is actually discontinuous or not and become all equal if X is continuous. Especially, one of them has the same statistical properties in any case. Efficiency is shown on a 2-D diffusive process as well as on the Boolean SATisfiability problem (SAT).

Inferring Evidence from Nested Sampling Data via Information Field Theory

“…This estimate is derived from the fact that the expectation value of the logarithmic volume changes is ⟨ln t i ⟩ = −1/N. However, this estimate does not take the uncertainties in the evidence calculation [5] into account and differs from unbiased approaches introduced and analysed in [6][7][8]. In any case, the imprecise knowledge of the prior volume introduces probing noise that can potentially hinder the accurate calculation of the evidence.…”

“…) assuming X 0 = 1 and X m+1 = 0. Adding dead points to their set and adjusting the evidence accordingly continues until the remaining live points occupy a tiny prior volume that would contribute little to the weighted sum in Equation (6).…”

A rare event approach to high-dimensional approximate Bayesian computation

“…It would be useful to investigate more general solutions from the rare event literature (e.g. Walter, 2015). Despite these potential issues, RE-ABC can perform well with discrete data in practice, for example in the binned data model of Section 5.3.…”