ANalyzing the effect of introducing a kurtosis parameter in gauSsian bayesian networks

,P .Main1*, H. Navarro2 
1Complutense University, Spain; 2UNED, Spain 
pmain@mat.ucm.es

Gaussian Bayesian Networks are graphical models that represent the dependence structure of a multivariate normal random variable with a DAG (directed acyclic graph); see, for example, Cowell et al. [2], Shachter and Kenley [4]. This qualitative aspect of the model has to be put with a quantitative part made up by the conditional distributions of each node given its parents in the graph, which are the preceding nodes. These conditional distributions are also normal distributions and the calculations are easily implemented. 
On the other hand the multivariate exponential power distribution (Gómez et al. [3]) is a family depending on a kurtosis parameter that goes from leptokurtic to platykurtic distributions with the normal as a mesokurtic distribution. This distribution is a generalization of the univariate distribution that was considered in Box & Tiao [1] for robustness studies in Bayesian inference and used later with this purpose in many situations. The conditional distributions are elliptical and can be also easily handled for calculations.
The problem of uncertainty about assumption of normality is very common in applications. Thus a sensitivity analysis of the non-normality effect in our conclusions is necessary. The kurtosis parameter of the multivariate exponential power distributions becomes the main tool to deal with deviations from the normal distribution.
The output  in Gaussian Bayesian Networks is usually the conditional distribution of the unknown variables of interest given some evidence, that is some known values for the rest of variables. Therefore a more general model can be considered using the multivariate exponential power distribution to describe the joint distribution of the Bayesian Network, with a kurtosis parameter reflecting deviations from the normal distribution. The sensitivity of the output to this perturbation is analyzed using the Kullback-Leibler divergence measure, studying the effect of slight non-normality in different ways. Some applications are also given.

References

[1] Box, G.E.P. and Tiao, G.C., Bayesian Inference in Statistical Analysis, Wiley, 1992.

[2] Cowell R G, Dawid A P, Lauritzen S L, and Spiegelhalter D J: Probabilistic Networks and
Expert Systems. Springer–Verlag, 1999.

[3] Gómez, E., Gómez-Villegas, M. A., Marín, J. M., A multivariate generalization of the power exponential family of distributions. Comm. Statist. Theory Methods 27, 589 – 600 (1998)..

[4] Shachter R D and Kenley C: Gaussian influence diagrams. Management Science, 35, 527–550 (1989).