Metastatic cancer is a possible caue of a brain tumor and is also an explanation for an increased total serum calcium. In turn, either of these could cause a patient to fall into an occasional coma. Severe headache could also be explained by a brain tumor. a) Represent these causal links in a belief network. Let a stand for "metastatic cancer", I) stand for "increased total serum calcium", .2 stand for "brain tumor", (1' stand for "occasional coma", and 3 stand for "severe headaches". b) Give an example of an independence assumption that is implicit in this network. c) Suppose that the following probabilities are given: Pr[a) =.2 Pr[b | a) =.8 Pr[b | a') =.2 Pr[c | a) = .2 Pr[c| a') = .05 Pr[e | c) =.8 Pr[e| c'] =.6 Prld l I") =-8 Pr[d| 'b,c) =.8 Prld l 5'67 = '8 Pr[d| 'b,c-) = .05 and assume that it is also given that some patient is suffering from severe headaches but has not fallen into a coma. Calculate the joint probabilities for the eight remaining possibilities (that is, according to whether a, b, and c are true or false}. cl) According to the numbers given, the a priori probability that the patient has metastatic cancer is _2 . Given that the patient is suffering from severe headaches but has not fallen into a coma, are we now more or less inclined to believe that the patient has cancer? Justify your answer. 3. [20 points] Imagine that we have a collection of blocks on a table and a robot arm capable of picking up blocks and putting them elsewhere, as shown in Figure 1. Figure 1: The Blocks World We assume that C - - the robot arm can hold at most one block at a assume that the pick up a block if block on top of time. We also robot can only there is no other