Belief Function Example

An Example using Belief Functions

Models using belief function instead of probabilities look almost the same in Graphical-Belief, but there are a few small differences. To study the problem, we look at a simple fault tree defined in Dempster and Kong [1988]. Performing the calculations we find that belief of system failure is .02 and the plausibility of system failure is .05.

Figure 1. Graphical Model Simple Machine after conditioning on system failure. Note that bars on the sides of nodes now show both plausibilities (white bar) and beliefs (black bar).

Figure 1 shows the model after conditioning on system failure. Once again we can use colors to track the most likely failure causes. Figure 2 shows a probe on the failure cause variable X2. It gives both the probability and belief values both before and after conditioning on system failure.

Figure 2. Probe showing failure cause X2 plausibilities and beliefs after conditioning on system failure.

Because we get an interval instead of a single number with the belief function models, it can be more difficult to make a decision. For example, suppose that our specifications for the machine said that the failure probability must be less than .025. This is in the middle of our calculated range, what should we do?

This is a weak decision: the model is too weak to make a proper decision. One solution is to simply ignore the ambiguity and to arbitrarily pick one of the limits or a point in between, if the loss from making a non-optimal decision is small that is certainly the best approach. Another solution is go back and refine the model, replacing imprecise judgements with more precise judgements. Almond[1995] explains this approach, it is closely related to the idea of test selection.

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