Differentiability and Approximation of Probability Functions under Gaussian Mixture Models

In this work, we study probability functions associated with Gaussian mixture models. Our primary focus is on extending the use of spherical radial decomposition for multivariate Gaussian random vectors to the context of Gaussian mixture models, which are not inherently spherical, but conditionally so. Specifically, the conditional probability distribution, given a random parameter of the random vector, follows a Gaussian distribution, which allows us to rewrite the probability function as a tractable integrated Gaussian mixture. This assumption, together with spherical radial decomposition for Gaussian random vectors, enables us to represent the probability function as an integral over the Euclidean sphere. Using this representation, we establish sufficient conditions to ensure the differentiability of the probability function and provide an integral representation of its gradient. Furthermore, we approximate the probability function using random sampling over the parameter space and the Euclidean sphere. Finally, we present a numerical example that illustrates the advantages of this approach over classical approximations based on random vector sampling.