Analyzing the effect of introducing a kurtosis parameter in Gaussian Bayesian networks

Gaussian Bayesian networks are graphical models that represent the dependence structure of a multivariate normal random variable with a directed acyclic graph (DAG). In Gaussian Bayesian networks the output is usually the conditional distribution of some unknown variables of interest given a set of evidential nodes whose values are known. The problem of uncertainty about the assumption of normality is very common in applications. Thus a sensitivity analysis of the non-normality effect in our conclusions could be necessary. The aspect of non-normality to be considered is the tail behavior. In this line, the multivariate exponential power distribution is a family depending on a kurtosis parameter that goes from a leptokurtic to a platykurtic distribution with the normal as a mesokurtic distribution. Therefore a more general model can be considered using the multivariate exponential power distribution to describe the joint distribution of a Bayesian network, with a kurtosis parameter reflecting deviations from the normal distribution. The sensitivity of the conclusions to this perturbation is analyzed using the Kullback–Leibler divergence measure that provides an interesting formula to evaluate the effect.