Consider a goat that may be infected with a disease that can possibly be detecte

Question

Consider a goat that may be infected with a disease that can possibly be detected by performing a milk test. The test is performed on five consecutive days, leading to five outcomes. We want to determine the state of the goat’s infection over these days given the test outcomes. The prior probability of an infection on day one is 1/10,000; the test false positive rate is 5/1,000; and its false negative rate is 1/1,000. Moreover, the state of infection at a given day depends only on its state at the previous day. In particular, the probability of a new infection on a given day is 2/10,000, while the probability that an infection would persist to the next day is 7/10. A.Describe a Bayesian network and a corresponding query that solves this problem. B.What is the most likely state of the goat’s infection over the five days given the following test outcomes: a.positive, positive, positive, negative, positive b.positive, negative, negative, positive, negative c.positive, negative, negative, positive, positive

Explanation / Answer

We can use Bayes Theorem which relates the conditional probability of events. The base definition is that the probability of an event A happening given B is the ratio of probabilities of A alone and B alone times the probability that B occur given A.
P(A|B)=P(B|A)A/B

In our example on the denominator the probability that someone tests positive is the sum of the probability they test positive and the disease is Present plus the probability they test positive and the disease is Absent so we have:

Pr(P|+)=Pr(+|P)Pr(P) / Pr(+|P)Pr(P)+Pr(+|A)Pr(A)

where P: disease is present, A: disease is absent, and +: positive test result

So we have:
Pr(P|+) 0.0001 / (0.0001+0.0050.9801) = 0.02 (approx)

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