تجزیه و تحلیل حساسیت در شبکه های بیزی گوسی با استفاده از یک روش نمادین عددی

The paper discusses the problem of sensitivity analysis in Gaussian Bayesian networks. The algebraic structure of the conditional means and variances, as rational functions involving linear and quadratic functions of the parameters, are used to simplify the sensitivity analysis. In particular the probabilities of conditional variables exceeding given values and related probabilities are analyzed. Two examples of application are used to illustrate all the concepts and methods.

Sensitivity analysis is becoming an important and popular area of work. Plain outputs of mathematical models are often insufficient for practical decision-making; the outputs must be further evaluated before a solid ground for decision-making has been established. To evaluate a model output, the model can be exposed to sensitivity analysis, indicating how sensitive the output is to change in values of model parameters [4], [5], [6], [7] and [8]. In some cases, the choice of parameter values has an extreme influence on the model outputs. For example, it is well known how sensitive the distributional assumptions and parameter values are to tail distributions (see [2], [9], [10] and [14]). If this influence is neglected, the consequences can be disastrous. Thus, sensitivity analysis plays a very important role in model validation and model output evaluation. Laskey [17] seems to be the first to address the complexity of sensitivity analysis of Bayesian networks. She introduced a method for computing the partial derivative of a posterior marginal probability with respect to a given parameter. Castillo et al. [5] and [4] show that the function expressing the posterior probability is a quotient of linear functions in the parameters and the evidence values in the discrete case, and of the means, variances, and evidence values, but covariances can appear squared. This discovery allows simplification of sensitivity analysis and makes it computationally efficient (see, for example, Ref. [16] or Ref. [12]). In this paper we address the problem of sensitivity analysis in Gaussian Bayesian networks and show how changes in the parameter and evidence values influence marginal and conditional probabilities given the evidence. The paper is structured as follows. In Section 2 we briefly review Gaussian Bayesian networks and introduce our working example. In Section 3 we discuss how to perform exact propagation in Gaussian Bayesian networks. Section 4 is devoted to symbolic propagation. Section 5 analyses the sensitivity problem. Section 6 presents the damage of concrete structures example. Finally, in Section 7 we make some concluding remarks.

From the previous sections, the following conclusions can be obtained: 1. Sensitivity analysis in Gaussian Bayesian networks is greatly simplified due to the knowledge of the algebraic structure of conditional means and variances. 2. The algebraic structure of any conditional mean or variance is a rational function of the parameters. 3. The degrees of the numerator and denominator polynomials in the parameters can be immediately identified, as soon as the parameter or evidence value is defined. 4. Closed expressions for the partial derivatives of probabilities of the form and with respect to the parameters, or evidence values, can be obtained. 5. Much more that sensitivity measures can be obtained. In fact, closed formulas for the probabilities and as a function of the desired parameter, or evidence value, can be obtained.