بررسی اثر معرفی یک پارامتر درجه اوج در یک نمودار اماری در شبکه های بیزی گاوسی

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.

A Bayesian network (BN) represents a multivariate random variable by means of directed acyclic graphs (DAG) [1] and [2]. This representation, that describes the dependence structure of the variables, has an intuitive interpretation even for complex models. On the other hand, the quantitative part of the network is given by the corresponding conditional distributions of each variable conditioned by its parents in the graph. Both parts describing the BN can also be used to efficiently make the calculations required for inference about the variables considered. The most important aspect is the evaluation of the effect that evidence about some variables causes to the remainder variables; it is known as evidence propagation. Some algorithms have been implemented giving tools available for inference in BNs [1], [2], [3] and [4]. These facilities explain the extensive use of BNs in applications [5], [6] and [7]. In particular they have been found useful as a framework for reliability analysis [8]. In Gaussian BNs (GBN), a joint multivariate normal distribution, N(μ,Σ)N(μ,Σ), is considered. This is usually because the conditionals are also normal distributions and the evidence propagation is obtained directly. However, in practice many problems do not fit this joint model distribution. Thus, in a generalization study, the multivariate exponential power (MEP) distribution [9] could be used. Now, the conditionals are elliptically distributed and the evidence propagation algorithm does not seem to be as direct as the normal case. Nevertheless, for a given GBN, a sensitivity analysis could provide information about the effect of deviations from normality in the tails. The output to be considered is, as usual in GBN, the conditional distribution of some variables of interest given known values of the evidential variables. Then, we study the difference between the outputs when varying the kurtosis parameter of the MEP family of distributions by means of the Kullback–Leibler (KL) divergence measure [10]. The MEP is a family that extends the corresponding univariate distribution [11]. It has been frequently used for robustness studies because it depends on a kurtosis parameter, ββ, that goes from zero to infinity, with β=1β=1 for the normal distribution, View the MathML sourceβ=12 for the Laplace distribution and β→∞β→∞ converging to a uniform distribution. The multivariate extension belongs to the large family of elliptical distributions that includes the multivariate normal. The elliptical continuous distributions are those whose density functions are constant over ellipsoids; this fact explains that many of the nice properties of normal distributions are maintained. The KL divergence is an asymmetric dissimilarity measure between two probability distributions from an information-theoretic basis. We use it to evaluate the difference between the outputs for a joint distribution with β=1β=1 and with some other β∈(0,∞)β∈(0,∞). The usage of a divergence measure gives a full comparison of the two distributions, not just in some characteristics such as the mean or the variance. With this procedure, if a multivariate normal distribution is used to handle the network but there are some doubts about this model, the sensitivity analysis for the BN reflects the influence of some model deviations on the network output. Sensitivity analysis in BNs is a developing area of work because its significance in practical situations where the model assumptions are doubtful. Most of the analyses deal with discrete networks [12] and [13] evaluating the perturbation in the parameters that describe the network. In this respect some sensitivity analyses are studied for GBNs [14] focusing on the effect on the conditional distribution parameters after the evidence propagation. Some other paper [15] uses the KL divergence measure to perform sensitivity analyses in GBNs for perturbations on the parameters of the joint normal distribution of the GBN. The paper is structured as follows. In Section 2 we set up notation and terminology and review some of the results to be used. In Section 3 our main result is stated and proved. Finally, in Section 4 some relevant examples are analyzed with the proposed methodology and in Section 5 some concluding remarks are given.

In GBN the possibility of performing a sensitivity analysis is helpful if there are some doubts about the normality of the joint distribution. We have proposed a methodology to deal with a wider class of distributions, including the normal, that considers slightly different tail behavior than the normal tail. This is the MEP family of distributions with a kurtosis parameter to give the possible forms of the distributions. On the other hand, the output in GBN to be analyzed corresponds to the conditional distribution of some variables of interest in the network given some known values of the evidential variables. Therefore in a sensitivity analysis, we have to compare the output with a normal model and with some other member of the MEP family. Our method uses the KL divergence measure between both outputs to make comparisons. With our approach the global comparison of the distributions is possible comparing at the same time all the characteristics of the distributions. Also, the usage of the MEP family has some benefits because we can use distributions very similar to the normal but, at the same time, they can reflect different tail behavior, easily characterized by a kurtosis parameter. Thus, a formula including all the elements that influence on the output when some normal deviated model is given. It is also calculated for some particular cases showing different situations that can be used to analyze the robustness of the networks.