We have a bag of three biased coins A, B, and C. When flipped, they have probabilities of coming up heads of 0.2, 0.5, and 0.8, respectively. One coin is drawn randomly from the bag (with equal likelihood of drawing each of the three coins), and then the coin is flipped three times to generate the outcomes X1 , X2 , and X3 .

a. Draw the Bayesian network corresponding to this setup and define the necessary CPTs.
b. Calculate which coin was most likely to have been drawn from the bag if the observed flips come out heads twice and tails once.

PART (a)

Let’s consider C as a random variable indicating the coin that we drew from a, b, c.  

For network, C would be at the root and X1, X2 and X3 would be considered as children.

The Conditional Probability Table (CPT) for C:

CP (C)

a1/3

b1/3

c1/3

The CPT for X1:

CX1          P (C)

aheads  0.2

bheads  0.6

cheads  0.8

PART (b)

According to the question, the following sequence is considered where P is the probability:

P(C|2 heads, 1 tails) = P (2 heads, 1 tails|C)P(C) / P(2 heads, 1 tails)

                                P (2 heads, 1 tails|C)P(C)

                                P (2 heads, 1 tails|C)

In the second line, we find that the constant of proportionality 1/P (2 heads, 1 tails) is independent of C, and P(C) is also independent of C in the last line, so, hypothetically, it is equal to 1/3.

In part a, we can see that X1, X2, and X3 are conditionally independent given C, therefore, for instance

P (X1 = tails, X2 = heads, X3 = heads | C = a)

   = P (X1 = tails | C = a) P (X2 = heads | C = a) P (X3 = heads | C = a)

   = 0.8 × 0.2 × 0.2 = 0.032

We would get the same probability for any order of 2 heads and 1 tails because the CPTs for each coin (a,b,c) are the same. Therefore, we have three such orders:

P(2 heads, 1 tails | C = a) = 3 × 0.032 = 0.096

P(2 heads, 1 tails | C = b) = 0.432

P(2 heads, 1 tails | C = c ) = 0.384

From the above results, it is concluded that coin b is most likely to have been drawn from the bag in the condition that is mentioned in the question.

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