Topic: Bayesian Networks
Related Reading: Ch. 14
Due: Thursday, 14 November 2013, 2:15pm

Please make sure you adhere to the policies on academic integrity.

1. (20 points)

   We have a bag of three biased coins, a, b, and c, with probabilities of coming up heads of 20%, 60%, and 80%, 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, X₃.

   1. Draw a (simple) Bayesian network corresponding the this setup and define the necessary CPTs.
   2. Calculate the conditional probability P(Coin | X₁=head, X₂=head, X₃=tail). That is, although there was a prior probability that each coin was picked with probability 1/3, what is the posterior probability that the coin is a, b, or c. (Hint: use Bayes' Theorem)
2. (30 points)

   Let H_x be a random variable denoting the handedness of a person x, with possible values l for x being left-handed and r for x being right-handed. A popular hypothesis is that left- or right-handedness is inherited by a simple mechanism; that is, perhaps there is a gene G_x, also with values l or r, and actual handedness turns out the same (with some probability s) as the gene that the person possesses. Furthermore, the gene itself is equally likely to be inherited from either of a person's parents, with a small non-zero probability m of a random mutation flipping the handedness.

   Consider the following three Bayesian Networks.

   (1)	(2)	(3)

   1. Which of the three networks claims that P(G_father, G_mother, G_child) = P(G_father) P(G_mother) P(G_child) ?
   2. Which of the three networks make independence claims that are consistent with the hypothesis about the inheritance of handedness?
   3. Which of the three networks is the best description of the hypothesis?
   4. Write down the CPT for the G_child node in network (1), in terms of s and m.
   5. Suppose that P(G_mother = l) = P(G_father = l) = q. In network (1), derive an expression for P(G_child = l) in terms of m and q only, by conditioning on its parent nodes.
   6. Under conditions of genetic equilibrium, we expect the distribution of genes to be the same across generations. Use this to calculate the value of q, and given what you know about handedness in humans, explain why the hypothesis described at the beginning of the question must be wrong.