Mixture density modeling, Kullback-Leibler divergence, and differential log-likelihood

A new log-likelihood (LL) based metric for goodness-of-fit testing and monitoring unsupervised learning of mixture densities is introduced, called differential LL. We develop the metric in the case of a Gaussian kernel fitted to a Gaussian distribution. We suggest a possible differential LL learning strategy, show the formal link with the Kullback-Leibler divergence and the quantization error, and introduce a Gaussian factorial distribution approximation by subspaces.