Efficient estimation of multiple expectations with the same sample by adaptive importance sampling and control variates

Abstract

Some classical uncertainty quantification problems require the estimation of multiple expectations. Estimating all of them accurately is crucial and can have a major impact on the analysis to perform, and standard existing Monte Carlo methods can be costly to do so. We propose here a new procedure based on importance sampling and control variates for estimating more efficiently multiple expectations with the same sample. We first show that there exists a family of optimal estimators combining both importance sampling and control variates, which however cannot be used in practice because they require the knowledge of the values of the expectations to estimate. Motivated by the form of these optimal estimators and some interesting properties, we therefore propose an adaptive algorithm. The general idea is to adaptively update the parameters of the estimators for approaching the optimal ones. We suggest then a quantitative stopping criterion that exploits the trade-off between approaching these optimal parameters and having a sufficient budget left. This left budget is then used to draw a new independent sample from the final sampling distribution, allowing to get unbiased estimators of the expectations. We show how to apply our procedure to sensitivity analysis, by estimating Sobol’ indices and quantifying the impact of the input distributions. Finally, realistic test cases show the practical interest of the proposed algorithm, and its significant improvement over estimating the expectations separately.

Appendix A: Equivalence between both optimization problem

Let us prove that the optimization problem in Eq. (18) is equivalent to the one in Eq. (21). Consider a sequence $\varvec{\alpha }\in S_J$, a family of IS auxiliary distributions $\left( g_{\varvec{\lambda }_j}\right) _{j\in [\![1,J]\!]}$, a family of control parameters $\left( \beta _j\right) _{j\in [\![1,J]\!]}\in {\mathbb {R}}^J$ and a family of positive weights $\left( w_j\right) _{j\in [\![1,J]\!]}\in {\mathbb {R}}_+^J$.

For any $j\in [\![1,J]\!]$ and any IS auxiliary distribution h, we have:

$$\begin{aligned}&{\mathbb {V}}_{g_{\varvec{\alpha }}}\left( \dfrac{\phi _j\left( {\textbf{X}}\right) f_j\left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) }{g_{\varvec{\alpha }}\left( {\textbf{X}}\right) } \right) \\&\quad = {\mathbb {E}}_{g_{\varvec{\alpha }}}\left[ \left( \dfrac{\phi _j\left( {\textbf{X}}\right) f_j\left( {\textbf{X}} \right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) }{g_{\varvec{\alpha }}\left( {\textbf{X}} \right) }\right) ^2\right] \\&\quad - {\mathbb {E}}_{g_{\varvec{\alpha }}}\left( \dfrac{\phi _j\left( {\textbf{X}} \right) f_j\left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) }{g_{\varvec{\alpha }}\left( {\textbf{X}}\right) }\right) ^2\\&\quad = {\mathbb {E}}_{g_{\varvec{\alpha }}}\left[ \dfrac{\left( \phi _j\left( {\textbf{X}}\right) f_j \left( {\textbf{X}} \right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) \right) ^2}{g_{\varvec{\alpha }} \left( {\textbf{X}}\right) ^2}\right] \\&\quad - \underbrace{{\mathbb {E}}_{f_j}\left( \phi _j\left( {\textbf{X}}\right) -\dfrac{\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) }{f_j\left( {\textbf{X}}\right) }\right) ^2} _{= c_j \text { independent of } \varvec{\alpha }}\\&= {\mathbb {E}}_{h}\left[ \dfrac{\left( \phi _j\left( {\textbf{X}}\right) f_j\left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) \right) ^2}{g_{\varvec{\alpha }} \left( {\textbf{X}}\right) h\left( {\textbf{X}}\right) }\right] - c_j. \end{aligned}$$

Therefore, we have:

$$\begin{aligned}&\sum _{j=1}^J w_j{\mathbb {V}}_{g_{\varvec{\alpha }}}\left( \dfrac{\phi _j\left( {\textbf{X}} \right) f_j\left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) }{g_{\varvec{\alpha }}\left( {\textbf{X}}\right) }\right) \\&\quad =\sum _{j=1}^J w_j\left( {\mathbb {E}}_{h}\left[ \dfrac{\left( \phi _j\left( {\textbf{X}}\right) f_j \left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) \right) ^2}{g_{\varvec{\alpha }}\left( {\textbf{X}}\right) h\left( {\textbf{X}}\right) }\right] - c_j\right) \\&\quad = \sum _{j=1}^J w_j{\mathbb {E}}_{h}\left[ \dfrac{\left( \phi _j\left( {\textbf{X}}\right) f_j\left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) \right) ^2}{g_{\varvec{\alpha }}\left( {\textbf{X}}\right) h\left( {\textbf{X}}\right) }\right] - \sum _{j=1}^J w_jc_j\\&\quad = {\mathbb {E}}_{h}\left[ \dfrac{\sum _{j=1}^Jw_j\left( \phi _j\left( {\textbf{X}}\right) f_j\left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) \right) ^2}{g_{\varvec{\alpha }}\left( {\textbf{X}} \right) h\left( {\textbf{X}}\right) }\right] \\&\quad - \sum _{j=1}^J w_jc_j. \end{aligned}$$

Since the term $\sum _{j=1}^J w_jc_j$ does not depend on the sequence $\varvec{\alpha }$, minimizing $\sum _{j=1}^J w_j{\mathbb {V}}_{g_{\varvec{\alpha }}}\left( \dfrac{\phi _j\left( {\textbf{X}} \right) f_j\left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) }{g_{\varvec{\alpha }}\left( {\textbf{X}}\right) }\right)$ w.r.t. $\varvec{\alpha }$ is then equivalent to minimize ${\mathbb {E}}_{h}\left[ \dfrac{\sum _{j=1}^Jw_j\left( \phi _j\left( {\textbf{X}} \right) f_j\left( {\textbf{X}}\right) -\beta _j g_{\varvec{\lambda }_j}\left( {\textbf{X}}\right) \right) ^2}{g_{\varvec{\alpha }} \left( {\textbf{X}}\right) h\left( {\textbf{X}}\right) }\right]$ w.r.t. $\varvec{\alpha }$. As a conclusion, both optimization problems in Eqs. (18) and (21) are equivalent.