In this work, we evaluate the sensitivity of Gaussian Bayesian networks to perturbations or uncertainties in the regression coefficients of the network arcs and the conditional distributions of the variables. The Kullback–Leibler divergence measure is used to compare the original network to its perturbation. By setting the regression coefficients to zero or non-zero values, the proposed method can remove or add arcs, making it possible to compare different network structures. The methodology is implemented with some case studies.
Highlights
► A new methodology to deal with perturbations in the conditional specification of Gaussian Bayesian networks is proposed. ► We can remove or add arcs, making it possible to compare different network structures. ► Some practical examples and a case study in metrology demonstrate the feasibility of the procedure.
A Bayesian network (BN) is a probabilistic model of causal interactions between a set of variables, where the joint probability distribution is described in graphical terms. Probabilistic networks have become an increasingly popular paradigm for reasoning through uncertain, complex models in a variety of situations, including AI, medical diagnosis and data mining (Kjærulff & Madsen, 2008).
This model consists of two parts: one qualitative and the other quantitative. Its qualitative aspect is a directed, acyclic graph (DAG), with nodes and arcs that represent a set of variables and their relationships respectively. Based on the dependence structure depicted in the graph, the joint distribution of the variables can be factorized in terms of the univariate conditional distributions of each variable given its parents in the DAG. These distributions constitute the quantitative portion of the model.
Building a BN is a difficult task, because all of the individual distributions and relationships between variables need to be correctly specified. Expert knowledge is essential to fix the dependence structure among variables of the network and to determine a large set of parameters. Databases can aid the process, but provide incomplete data and only partial knowledge of the domain. Thus, any assessments obtained using only databases are inevitably inaccurate (van der Gaag, Renooij, & Coupe, 2007).
The present research is restricted to a subclass of BNs known as Gaussian Bayesian networks (GBNs). The quantitative portion of a GBN consists of a univariate normal distribution for each variable given its parents in the DAG. Also, the joint probability distribution of the model is constrained to be a multivariate normal distribution.
For each variable Xi, the experts have to provide its mean, the regression coefficients between Xi and each parent Xj ∈ pa(Xi) ⊂ {X1, … , Xi−1}, and the conditional variance of Xi given its parents. This specification is easy for experts, because they only have to describe univariate distributions. Moreover, the arcs in the DAG can be expressed in terms of the regression coefficients.
Our interest in this paper is the sensitivity of GBNs defined by these parameters. This subject has not been frequently treated in the literature, because sensitivity analyses usually perturb the joint parameters instead of the conditional parameters. However, it is easy to model the presence or absence of arcs by adopting regression coefficients different from or equal to zero. Thus, it is also possible to study the effect of changes in the qualitative part of the network. An objective evaluation of this effect may also reveal that a simpler DAG structure yields equivalent results.
In Section 1 we define GBNs, present some general concepts, and introduce a working example. In Section 2 we describe the methodology used to study the sensitivity of the GBN and calculate the sensitivity of our example. In Section 3 we vary the network structure and present a metrology example: the calibration of an electronic level using a sine table. The paper ends with some conclusions and suggestions for further research.
This paper contributes to the problem of sensitivity analysis in GBNs in three ways. First, it describes how to characterize uncertainty in the conditional specifications of a GBN. Second, it explains how to analyze the network’s sensitivity to perturbations in the network parameters (means, conditional variances, and the regression coefficients between a variable and its parents in the DAG). Third, it proposes a method of n-way sensitivity analysis that provides a global vision of the difference between the original network and its perturbation. We evaluate and discuss the proposed sensitivity analysis with an example GBN and several cases of uncertainty.
An important use of the model is evaluating the network’s sensitivity to structural variations. By replacing regression coefficients with zero or non-zero values, the new method can remove or add arcs in the DAG. The results are applied to a metrology case study: the calibration of an electronic level using a sine table.
Further research will focus on applying the previous results to establish the sensitivity of a network to specific nodes in the DAG. Another interesting extension to the model is including prior evidence on some of the variables; by this means we can evaluate the effect of perturbations on evidence propagation.
