This section contains a comprehensive example illustrating the application of the Q-DAG framework to diagnostic reasoning.

**Figure:** A simple belief network for car diagnosis.

Consider the car troubleshooting example depicted in Figure 14. For this simple case we want to determine the probability distribution for the fault node, given evidence on four sensors: the battery-, alternator-, fuel- and oil-sensors. Each sensor provides information about its corresponding system. The fault node defines five possible faults: normal, clogged-fuel-injector, dead-battery, short-circuit, and broken-fuel-pump.

If we denote the fault variable by *F*, and sensor variables by , then
we want to build a system that can compute the probability
for each fault and any evidence . These probabilities represent
an unnormalized probability distribution over the fault variable given
sensor readings. In a Q-DAG framework, realizing this diagnostic system involves
three steps: Q-DAG generation, reduction, and evaluation. The first two steps
are accomplished off-line, while the final step is performed on-line. We now
discuss each one of the steps in more detail.

Darwiche&Provan