多元高斯混合模型的惩罚极大似然
Penalized maximum likelihood for multivariate Gaussian mixture
摘要：
本文首先考虑利用多元高斯混合律估计多元随机过程分布的参数。标签的混合物被允许有一个一般的概率定律，这使得建模的时间结构的过程中的研究。我们推广了[Ridolfi 99]中的单变量高斯混合的情况，证明了当协方差矩阵之一接近非负定矩阵集的奇异边界时，似然是无界的，并且趋于无穷大。我们刻画了这些奇点的参数集。作为这个退化问题的解决方案，我们表明，惩罚的可能性由逆Wishart之前的协方差矩阵的结果，惩罚或最大后验标准是有界的。然后，存在正定矩阵优化这个准则可以保证。我们还表明，与修改后的EM程序或贝叶斯抽样方案，我们可以约束协方差矩阵属于一个特定的子类的协方差矩阵。最后，我们研究了源分离问题中的退化问题，其中参数奇异集的特征更为复杂。然而，我们发现，逆Wishart先验的协方差矩阵消除了这种情况下的退化。
英文摘要：
In this paper, we first consider the parameter estimation of a multivariate random process distribution using multivariate Gaussian mixture law. The labels of the mixture are allowed to have a general probability law which gives the possibility to modelize a temporal structure of the process under study. We generalize the case of univariate Gaussian mixture in [Ridolfi99] to show that the likelihood is unbounded and goes to infinity when one of the covariance matrices approaches the boundary of singularity of the non negative definite matrices set. We characterize the parameter set of these singularities. As a solution to this degeneracy problem, we show that the penalization of the likelihood by an Inverse Wishart prior on covariance matrices results to a penalized or maximum a posteriori criterion which is bounded. Then, the existence of positive definite matrices optimizing this criterion can be guaranteed. We also show that with a modified EM procedure or with a Bayesian sampling scheme, we can constrain covariance matrices to belong to a particular subclass of covariance matrices. Finally, we study degeneracies in the source separation problem where the characterization of parameter singularity set is more complex. We show, however, that Inverse Wishart prior on covariance matrices eliminates the degeneracies in this case too.
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