### CSE443 Reasoning Under Uncertainty

#### www.csse.monash.edu.au/~annn/443/Ex2.html

1. Orville the Juggler
2. Orville, the robot juggler, drops balls quite often when its battery is low. In previous trials, it has been determined that the probability that it will drop a ball when its battery is low is 0.9. On the other hand when its battery is not low, the probability that it drops a ball is only 0.01. The battery was recharged not so long ago, so there is only a 5% chance that the battery is low. A robot observor, with a somewhat unreliable vision system, reports on whether or not Orville has dropped the ball. This question involves constructing a belief network, containing only Boolean variables, to represent and draw inferences about whether the battery is low depending on how well Orville is juggling.

(a) Draw a belief network to represent the problem. Label the network nodes and indicate clearly the direction of the arcs between the nodes.

(b) Write down the probability tables showing where the information on how Orville's success is related to the battery level, and the robot observer's accuracy, are encoded in the network.

(c) Suppose the robot observer reports that Orville has dropped the ball. What effect does this have on your belief that the battery is low. What type of reasoning is being done?

3. Conditional Independence
4. Consider the following belief network for another medical diagnosis example, where B=Bronchitis, S=Smoker, C=Cough, X=Positive X-ray and L=Lung cancer. Suppose that the prior for a patient being a smoker is 0.25, and the prior for the patient having bronchitis (during winter in Melbourne!) is 0.05.

List the pairs of nodes that are conditionally independent in the following situations:

(a) There is no evidence for any of the nodes.
(b) The cancer node is set to true (and there is no other evidence).
(c) The smoker node is set to true (and there is no other evidence).
(d) The cough node is set to true (and there is no other evidence).

5. Variable ordering
6. (a) What variable ordering(s) would have been used to produce the above network using the network construction algorithm described in lectures (Lecture 2, slide 10)?

(b) Given different variable orderings, what network structure would result from this algorithm?

7. D-separation.

(a) Find all the sets of nodes that d-separate X and Y in the graph below (not including either X or Y in such sets).

(b) Try to come up with a real-world scenario that might be modelled with such a network structure.