A patient is known to have contracted a rare discase which comes in two forms, represented...

 

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A patient is known to have contracted a rare discase which comes in two forms, represented by the values of a binary random variable X E (0,1). Symptoms of the discase are represented by the binary random variables Y E (0,1), and knowledge of the disease is summarized by the belief network: Y3 ... The conditional probability tables (CPTS) for this belief network are as follows. In the absence of evi- dence, both forms of the disease are equally likely, with prior probabilities: P(X=0) = P(X=1) =. In the first form of the disease (X = 0), all the symptoms are uniformly likely to be observed, with P(Y=0|X =0) = for all k. By contrast, in the second form of the disease (X =1), the first symptom occurs with probability one, %3D P(Y=1|X =1) = 1, while the kth symptom (with k22) occurs with probability P(Y=1|X 1) = /(k-1) %3D S(k) where the function f(k) is defined by S(k)= 24 +(-1)*. Suppose that on the kth day of the month, a test is done to determine whether the patient is exhibiting the Ath symptom, and that each such test returns a positive result. Thus, on the kth day, the doctor observes the patient with symptoms (Y 1, Y=1,...,Y=1}. Based on the cumulative evidence, the doctor makes a new diagnosis each day by computing the ratio: %3D P(X =1|Y=1, Y2 1,...,Y 1) P(X 0Y=1,Y=1,...,Y=1) If this ratio is greater than 1, the doctor diagnoses the patient with the X 1 form of the disease; otherwise, with the X 0 form. Compute the ratio r as a function of k. How does the doctor's diagnosis depend on the day of the month? Does the diagnosis become more or less certain as more symptoms are observed? Explain. A patient is known to have contracted a rare discase which comes in two forms, represented by the values of a binary random variable X E (0,1). Symptoms of the discase are represented by the binary random variables Y E (0,1), and knowledge of the disease is summarized by the belief network: Y3 ... The conditional probability tables (CPTS) for this belief network are as follows. In the absence of evi- dence, both forms of the disease are equally likely, with prior probabilities: P(X=0) = P(X=1) =. In the first form of the disease (X = 0), all the symptoms are uniformly likely to be observed, with P(Y=0|X =0) = for all k. By contrast, in the second form of the disease (X =1), the first symptom occurs with probability one, %3D P(Y=1|X =1) = 1, while the kth symptom (with k22) occurs with probability P(Y=1|X 1) = /(k-1) %3D S(k) where the function f(k) is defined by S(k)= 24 +(-1)*. Suppose that on the kth day of the month, a test is done to determine whether the patient is exhibiting the Ath symptom, and that each such test returns a positive result. Thus, on the kth day, the doctor observes the patient with symptoms (Y 1, Y=1,...,Y=1}. Based on the cumulative evidence, the doctor makes a new diagnosis each day by computing the ratio: %3D P(X =1|Y=1, Y2 1,...,Y 1) P(X 0Y=1,Y=1,...,Y=1) If this ratio is greater than 1, the doctor diagnoses the patient with the X 1 form of the disease; otherwise, with the X 0 form. Compute the ratio r as a function of k. How does the doctor's diagnosis depend on the day of the month? Does the diagnosis become more or less certain as more symptoms are observed? Explain.