623	RWTH Publication No: 976819        2023
TITLE	Taming numerical imprecision by adapting the KL divergence to negative probabilities
AUTHORS	Simon Pfahler, Peter Georg, Rudolf Schill, Maren Klever, Lars Grasedyck, Rainer Spang, Tilo Wettig
ABSTRACT	The Kullback-Leibler (KL) divergence is frequently used in data science. For discrete distributions on large state spaces, approximations of probability vectors may result in a few small negative entries, rendering the KL divergence undefined. We address this problem by introducing a parameterized family of substitute divergence measures, the shifted KL (sKL) divergence measures. Our approach is generic and does not increase the computational overhead. We show that the sKL divergence shares important theoretical properties with the KL divergence and discuss how its shift parameters should be chosen. If Gaussian noise is added to a probability vector, we prove that the average sKL divergence converges to the KL divergence for small enough noise. We also show that our method solves the problem of negative entries in an application from computational oncology, the optimization of Mutual Hazard Networks for cancer progression using tensor-train approximations.
KEYWORDS	Kullback-Leibler divergence, Approximate Bayesian computation, Statistical optimization, Mutual Hazard Networks, Tensor trains