2010/12/05 by Nicolás Bousquet, Bousquet, Nicolas
Decision Sciences · Computer Science · #Probabilistic and Robust Engineering Design #Advanced Multi-Objective Optimization Algorithms #Advanced Statistical Process Monitoring
paper · pdf · doi:10.48550/arxiv.1012.1042
The problem of estimating the probability p=P(g(X<0) is considered when X\nrepresents a multivariate stochastic input of a monotone function g. First, a\nheuristic method to bound p is formally described, involving a specialized\ndesign of numerical experiments. Then a statistical estimation of p is\nconsidered based on a sequential stochastic exploration of the input space. A\nmaximum likelihood estimator of p based on successive dependent Bernoulli data\nis defined and its theoretical convergence properties are studied. Under\nintuitive or mild conditions, the estimation is faster and more robust than the\ntraditional Monte Carlo approach, therefore adapted to time-consuming computer\ncodes g. The main result of the paper is related to the variance of the\nestimator. It appears as a new baseline measure of efficiency under monotone\nconstraints, which could play a similar role to the usual Monte Carlo estimator\nvariance in unconstrained frameworks. Furthermore the bias of the estimator is\nshown to be corrigible via bootstrap heuristics. The behavior of the method is\nillustrated by numerical tests led on a class of toy examples and a more\nrealistic hydraulic case-study.\n Keywords : monotone function, deterministic computer codes, Monte Carlo\nacceleration, failure probability\n