Upon varying parameters in a sensitivity analysis of a Bayesian network, the standard approach is to co-vary the parameters from the same conditional distribution such that their proportions remain the same. Alternative co-variation schemes are, however, possible. In this paper we investigate the properties of the standard proportional co-variation and introduce two alternative schemes: uniform and order-preserving co-variation. We theoretically investigate the effects of using alternative co-variation schemes on the so-called sensitivity function, and conclude that its general form remains the same under any linear co-variation scheme. In addition, we generalise the CD-distance for bounding global belief change to explicitly include the co-variation scheme under consideration. We prove a tight lower bound on this distance for parameter changes in single conditional probability tables.
Sensitivity analysis is a general technique for studying the effects of parameter changes on the output of a mathematical model. In the context of Bayesian networks the output of interest can be any probability computed from the network; the parameters to which changes are applied consist of one or more probabilities from the network's conditional probability tables. The results of a sensitivity analysis can be captured in detail by means of a sensitivity function, describing an output probability of interest as a function of one or more parameter probabilities [1]. More global effects of parameter changes can be described by the CD-distance [2]. The CD-distance is a measure for bounding probabilistic belief change and complements the sensitivity function by giving insight in the effect of parameter changes on the global joint distribution, rather than on a specific (posterior) output probability of interest.
Upon varying a probability from a (conditional) distribution, the remaining probabilities from the same distribution need to be co-varied. The proportional scheme has been adopted as the standard scheme for co-variation in Bayesian networks, and various algorithms and properties associated with sensitivity analysis build upon this scheme [1], [2], [3], [4] and [5]. For example, the known standard form of the sensitivity function is based on proportional co-variation [1]. The proportional co-variation scheme, however, is one of numerous alternatives for co-varying parameters from the same distribution. In this paper we investigate the properties of the proportional co-variation scheme and argue that the mere fact that it is the standard co-variation scheme used, does not imply that there are no situations in which alternative schemes may be more suitable. For example, we may want to prevent certain parameters from co-varying, or preserve some relation — such as the order — between parameters. We therefore introduce some alternative schemes, study their properties and compare sensitivity functions established under the different schemes.
A known property of the proportional co-variation scheme is that it is optimal in the context of varying a single parameter, in the sense that it minimises the CD-distance between the original and the new distribution [2]. It is as of yet unknown, however, if the proportional scheme is also optimal when multiple, independent parameters are varied. Moreover, the choice of the CD-distance as the measure to optimise in the context of sensitivity analysis is rather arbitrary: we could also be interested in minimising the well-known KL-divergence [6], or some other measure. Parameter changes that minimise KL-divergence do not necessarily minimise CD-distance, or vice versa [2]. In addition, to get a ‘worst case’ impression, we may want to perform our analyses in the context of large disturbances to the network, rather than minimal ones. In this paper we will therefore investigate exactly how both the sensitivity function and the CD-distance depend on the co-variation scheme used. We show that the general form of the sensitivity function is maintained as long as the co-variation scheme is linear in the parameter(s) varied. In addition, we generalise the CD-distance to arbitrary co-variation schemes, and prove that a previously suggested approximation of this distance is in fact a lower bound.
This paper is organised as follows. Section 2 provides preliminaries on Bayesian networks and sensitivity analysis. Section 3 discusses proportional co-variation, introduces alternatives and studies properties of the various schemes. In Section 4, we generalise the sensitivity function to arbitrary co-variation schemes; likewise, Section 5 generalises the CD-distance. The paper ends with conclusions and directions for future research in Section 6.
In this paper we have investigated various properties of the standard proportional co-variation scheme and argued that under some circumstances a different approach to co-variation may be preferred. We discussed several alternative co-variation schemes and their properties. In addition, we have generalised both sensitivity functions and CD-distance to cope with arbitrary co-variation schemes. We showed that the sensitivity function remains a rational function as long as a valid and linear co-variation scheme is used. In addition, we compared sensitivity functions and CD-distances under different co-variation schemes. Finally, we proved a lower bound on CD-distance for single CPTs.
The increased complexity of the formula for the CD-distance for full-single-CPT changes forestalls straightforward generalisation of the known optimality of the CD-distance for single parameter changes to full-single-CPT changes. The examples shown in Section 5.2, however, seem to support the claim that, for full-single-CPT changes, the CD-distance under proportional co-variation is also a lower bound on the CD-distance for other co-variation schemes. This claim, however, still needs to be formally proven.
Alternative co-variation schemes may be used to preserve important domain-dependent properties, such as known thresholds or relationships between parameters. Insight in the properties of different co-variation schemes helps in selecting the most suitable scheme for a sensitivity analysis of a specific Bayesian network. In addition, alternative co-variation schemes can be interesting to consider in the context of parameter tuning: it may very well be that single parameter or single CPT changes do not suffice to satisfy a certain constraint on an output probability under proportional co-variation, yet are able to satisfy the constraint under an alternative scheme.
