Bayesian networks [1+1* P]

Figure: Bayesian network.

a)

[1 P] Determine whether the following conditional independence statements hold for the Bayesian network illustrated in Fig. 5.

1. $(X3 \perp X7)$
2. $(X3 \perp X7) \vert X4$
3. $(X3 \perp X5)$
4. $(X3 \perp X5) \vert X7$

b)

[1* 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 $A$ .