To investigate the robustness of the output probabilities of a Bayesian network, a sensi tivity analysis can be performed. A one-way sensitivity analysis establishes, for each of the probability parameters of a network, a func tion expressing a posterior marginal proba bility of interest in terms of the parameter. Current methods for computing the coeffi cients in such a function rely on a large num ber of network evaluations. In this paper, we present a method that requires just a single outward propagation in a junction tree for es tablishing the coefficients in the functions for all possible parameters; in addition, an in ward propagation is required for processing evidence. Conversely, the method requires a single outward propagation for computing the coefficients in the functions expressing all possible posterior marginals in terms of a sin gle parameter. We extend these results to an n-way sensitivity analysis in which sets of pa rameters are studied. The robustness of the output probabilities of a Bayesian network can be investigated by performing a sensitivity analysis of the network. For mathemat ical models in general, sensitivity analysis serves to identify the effects of the inaccuracies in a model's pa rameters on its output (Morgan & Henrion 1990). For a Bayesian network, more specifically, performing a sensitivity analysis yields insight in the relation be tween the probability parameters of the network and its posterior marginals. The simplest type of sensi tivity analysis is a one-way analysis in which a single parameter is studied; a more general n-way analysis serves to investigate the joint effects of inaccuracies in In the brute-force approach to performing a one-way sensitivity analysis of a Bayesian network, each proba bility parameter of the network is varied systematically and the effect on the output probabilities of the net work is investigated. Performing a sensitivity analysis in this way requires thousands of network evaluations, or (full) propagations, and is, therefore, much too time consuming to be of any practical use. Laskey (1995) has been the first to address the compu tational complexity of sensitivity analysis of Bayesian networks. She has introduced a method for computing the partial derivative of a posterior marginal proba bility with respect to a parameter under study. Her method thus yields a first-order approximation of the effect of varying a single probability parameter on a posterior marginal. Compared to the brute-force ap proach, her method requires considerably less compu tational effort. The method, however, provides insight only in the effect of small variations of parameters; when larger variations are considered, the quality of the approximation may rapidly break down. The relation between a posterior marginal probability of interest and a parameter under study can be ex pressed through a simple mathematical function. The function expressing the posterior marginal is a quo tient of two linear functions in the parameter, as has been shown by Castillo et al. (1996) . Building upon this property, it suffices to establish the coefficients in this function to determine the effect of parameter variation. Castillo et al. (1997) and Coupe & van der Gaag (1998) have designed methods to this end. These methods require a single network evaluation for each coefficient to be established. Although these methods currently are the most efficient available, they rely on a large number of network evaluations and, as a con sequence, are infeasible for large realistic networks. In this paper, we present a new method for sensitivity analysis of Bayesian networks. Our method, like the
Van der Gaag et al. REF prove that sensitivity of an output variable in a Bayesian network with respect to each input CPT parameter can be computed in O(n) time (n is the size of the network) using techniques similar to junction tree belief propagation.
Making Sensitivity Analysis Computationally Efficient
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