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 only conditionally so. Specifically, the conditional probability distribution, given a random parameter of the random vector, follows a Gaussian distribution, allowing us to apply Bayesian analysis tools to the probability function. 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 and integral representation of its gradient. Furthermore, leveraging the Bayesian decomposition, we approximate the probability function using random sampling over the parameter space and the Euclidean sphere. Finally, we present numerical examples that illustrate the advantages of this approach over classical approximations based on random vector sampling.

翻译：本文研究了与高斯混合模型相关的概率函数。我们主要关注将多元高斯随机向量的球面径向分解方法推广至高斯混合模型的情境——此类模型本身并非球对称，而仅具有条件球对称性。具体而言，在给定随机向量参数条件下，其条件概率分布服从高斯分布，这使得我们可以将贝叶斯分析工具应用于概率函数。该假设结合高斯随机向量的球面径向分解，使我们能够将概率函数表示为欧几里得球面上的积分。基于此表示形式，我们建立了确保概率函数可微性的充分条件，并给出了其梯度的积分表示。进一步地，利用贝叶斯分解，我们通过参数空间与欧几里得球面上的随机采样对概率函数进行逼近。最后，我们通过数值算例展示了该方法相较于基于随机向量采样的经典逼近方法的优势。