Integral equalities and inequalities: a proxy-measure for multivariate sensitivity analysis

Weighted Poincar\'e-type and related inequalities provide upper bounds of the variance of functions. Their application in sensitivity analysis allows for quickly identifying the active inputs. Although the efficiency in prioritizing inputs depends on the upper bounds, the latter can be big, and therefore useless in practice. In this paper, an optimal weighted Poincar\'e-type inequality and gradient-based expression of the variance (integral equality) are studied for a wide class of probability measures. For a function $f : \mathbb{R} \to \mathbb{R}^n$, we show that $${\rm Var}_\mu(f) = \int_{\Omega \times \Omega} \nabla f (x) \nabla f (x')^T \frac{F(\min(x,\, x') - F(x)F(x')}{\rho(x) \rho(x')} \, d\mu(x) d\mu(x')\, ,$$ and $${\rm Var}_\mu(f) \preceq \frac{1}{2}\int_\Omega \nabla f(x) \nabla f(x)^T \frac{F(x) (1-F(x))}{[\rho(x)]^2}\, d\mu(x) \, ,$$ with ${\rm Var}_\mu(f) = \int_\Omega ff^T\, d\mu -\int_\Omega f \, d\mu \int_\Omega f^T\, d\mu$, $F$ and $\rho$ the distribution and the density functions, respectively. These results are generalized to cope with multivariate functions by making use of cross-partial derivatives, and they allow for proposing a new proxy-measure for multivariate sensitivity analysis, including Sobol' indices. Finally, the analytical and numerical tests show the relevance of our proxy-measure for identifying important inputs by improving the upper bounds from Poincar\'e inequalities.