Polynomial Approximations to the Sliced-Normal Density

Sliced-Normal (SN) distributions can be used to characterize strongly dependent random variables having multiple modes. Furthermore, their polynomial structure renders common uncertainty quantification tasks, such as identifying the true Most Probable Point (MPP) of failure, computationally tractable. However, the need to accurately estimate the normalization constant has limited their applicability to problems with a moderately large number of uncertain parameters. This note develops lower polynomial bounds and polynomial approximations to the SN joint density that enable computing such a constant analytically. Furthermore, we propose a polynomial class of distributions that exhibit not only the same versatility of the SNs but also the integrability benefits of the proposed approximations. This paper presents the mathematical framework supporting such developments along with easily reproducible examples.