Bayesian networks [2+2* P]

Figure: Graphical model for diagnosis example.

a)

[2 P] Consider the graphical model in Figure 4 for a medical diagnosis example, where $B$ = bronchitis, $S$ = smoker, $C$ = cough, $X$ = positive X-ray, and $L$ = lung cancer.

List the pairs of nodes that can be proven to be conditionally independent with the definition of d-separation, given the following evidence:

1)

[1/2 P] No evidence for any of the nodes.

2)

[1/2 P] The lung cancer node is set to true (and no other evidence).

3)

[1/2 P] The smoker node is set to true (and no other evidence).

4)

[1/2 P] The cough node is set to true (and no other evidence).

b)

[2* P] In your local nuclear power plant station, there is an alarm that senses when a temperature gauge exceeds a given threshold. The gauge measures the temperature of the core. Consider the Boolean variables $A$ (alarm sounds), $F_A$ (alarm is faulty), and $F_G$ (gauge is faulty) and the multivalued nodes $G$ (gauge reading) and $T$ (actual core temperature).

1. Draw a Bayesian network for this domain, given that the gauge is more likely to fail when the core temperature gets too high.
2. Suppose there are just two possible actual and measured temperatures, normal and high; the probability that the gauge gives the correct temperature is $x$ when it is working, but $y$ when it is faulty. Give the conditional probability table associated with $G$ .
3. Suppose the alarm works correctly unless it is faulty, in which case it never sounds. Give the conditional probability table associated with $A$ .