Lipschitz Parametrization of Probabilistic Graphical Models

We show that the log-likelihood of several probabilistic graphical models is Lipschitz continuous with respect to the �p-norm of the parameters. We discuss several implications of Lipschitz parametrization. We present an upper bound of the Kullback-Leibler divergence that allows understanding methods that penalize the �p-norm of differences of parameters as the minimization of that upper bound. The expected log-likelihood is lower bounded by the negative �p-norm, which allows understanding the generalization ability of probabilistic models. The exponential of the negative �p-norm is involved in the lower bound of the Bayes error rate, which shows that it is reasonable to use parameters as features in algorithms that rely on metric spaces (e.g. classification, dimensionality reduction, clustering). Our results do not rely on specific algorithms for learning the structure or parameters. We show preliminary results for activity recognition and temporal segmentation.