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## Bayesian Subset Simulation Tutorial

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Emmanuel Vazquez
Julien Bect

#### Abstract

We consider the problem of estimating a probability of failure $\alpha$, defined as the volume of the excursion set of a function $f: \mathbb{X} \subset \mathbb{R}^d \to \mathbb{R}$ above a given threshold, under a given probability measure on $\mathbb{X}$. In this talk, we present a tutorial about the BSS (Bayesian Subset Simulation) algorithm (Bect, Li and Vazquez, SIAM JUQ 2017), which combines 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. A key idea 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. BSS achieves significant savings in the number of function evaluations with respect to other Monte Carlo approaches. The BSS algorithm is implemented in the STK (Small Matlab/Octave Toolbox for Kriging).

#### Domains

Statistics [stat]

### Dates and versions

hal-02299402 , version 1 (27-09-2019)

### Licence

Attribution - NonCommercial - NoDerivatives - CC BY 4.0

### Identifiers

• HAL Id : hal-02299402 , version 1

### Cite

Emmanuel Vazquez, Julien Bect. Bayesian Subset Simulation Tutorial. SIAM Conference on Uncertainty Quantification (SIAM UQ 2018), Apr 2018, Garden Grove, Orange County, United States. ⟨hal-02299402⟩

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