Accurate and efficient representation of non-Gaussian state distributions is critical for estimation, control, and decision-making, especially when the system is highly nonlinear. This paper develops a mathematical framework for estimating non-Gaussian uncertainty via semi-nonparametric (SNP), or Gallant-Nychka, densities. SNP densities use squared probabilists' Hermite polynomials to guarantee nonnegative probability densities while representing non-Gaussian behavior. For uncertainty propagation, once fitted to propagated state samples, SNP densities enable direct analytical density and probability evaluations without needing to tune additional hyperparameters besides the polynomial order. To enable their application to nonlinear dynamical systems, we derive analytic marginal and cumulative distribution functions and develop a lossless convex formulation of the coefficient estimation problem along a given sign branch. The approach is demonstrated on the chaotic Lorenz system and compared against Gaussian mixture models and Edgeworth approximations. Results demonstrate that SNP densities can accurately represent non-Gaussian state distributions while enabling direct probability and quantile evaluation, making them a promising tool for density estimation and uncertainty quantification in nonlinear estimation and control.

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