We consider the problem of evaluating dynamic consistency in discrete time probabilistic filters that approximate stochastic system state densities with Gaussian mixtures. Dynamic consistency means that the estimated probability distributions correctly describe the actual uncertainties. As such, the problem of consistency testing naturally arises in applications with regards to estimator tuning and validation. However, due to the general complexity of the density functions involved, straightforward approaches for consistency testing of mixture-based estimators have remained challenging to define and implement. This paper derives a new exact result for Gaussian mixture consistency testing within the framework of normalized deviation squared (NDS) statistics. It is shown that NDS test statistics for generic multivariate Gaussian mixture models exactly follow mixtures of generalized chi-square distributions, for which efficient computational tools are available. The accuracy and utility of the resulting consistency tests are numerically demonstrated on static and dynamic mixture estimation examples.

翻译：本文研究离散时间概率滤波器中动态一致性的评估问题，该类滤波器利用高斯混合模型近似随机系统状态密度。动态一致性指估计概率分布能正确描述实际不确定性。因此，在估计器调优与验证等应用中，一致性检验问题自然产生。然而，由于所涉及密度函数的一般复杂性，针对基于混合模型的估计器定义并实施直接的一致性检验方法仍具挑战。本文在归一化偏差平方（NDS）统计量框架下，推导出高斯混合一致性检验的精确新结论。研究表明，通用多元高斯混合模型的NDS检验统计量精确服从广义卡方分布的混合分布，且存在高效计算工具。通过静态与动态混合估计实例，数值验证了所提一致性检验的准确性与实用性。