Which of these Bayesian networks are correct representations of the preceding information?

Directions: Complete questions 1-7 below.

⦁ (20pt) Consider the following Cryptarithmetic problems, where all letters represent a different digit and the resulting sum is correct. Write out all of the variables, domains and constraints of the problem.
⦁ SATURN
+ URANUS
= PLANETS

⦁ YES
SEND
ME
+ MORE
= MONEY

⦁ (20pt) Consider the following set of edges between nodes. Find a coloring using colors red, blue, and green such that no two adjacent nodes are assigned the same color.
{(a, b),(a, d),(b, c),(b, d),(b, g),(c, g),(d, e),(d, f),(d, g),(f, g)}.

⦁ Define a CSP for this problem. Clearly define the variables, domains, and constraints.
⦁ Draw the binary constraint graph for this CSP.
⦁ Find at least one solution to the CSP.

⦁ (20pt) Given the below full joint distribution, calculate the following:

⦁ P(toothache)
⦁ P(Cavity)
⦁ P(Toothache | cavity)
⦁ P(Cavity | toothache ∨ catch).

⦁ (20pt) We encounter a slot machine with three independent wheels, each producing one of the four symbols bar, bell, lemon, or cherry with equal probability. The slot machine has the following payout scheme for a bet of 1 coin (where “?” denotes that we don’t care what comes up for that wheel):
⦁ bar/bar/bar pays 20 coins
⦁ bell/bell/bell pays 15 coins
⦁ lemon/lemon/lemon pays 5 coins
⦁ cherry/cherry/cherry pays 3 coins
⦁ cherry/cherry/? pays 2 coins
⦁ cherry/?/? pays 1 coin
Compute the expected “payback” percentage of the machine. In other words, for each coin played, what is the expected coin return?

⦁ (20pt )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, and X3.
⦁ Draw the Bayesian network corresponding to this setup and define the necessary CPTs.
⦁ Calculate which coin was most likely to have been drawn from the bag if the observed flips come out heads twice and tails once.

⦁ (30pt) In your local nuclear power station, there is an alarm that senses when a temperature gauge exceeds a given threshold. The gauge measures the temperature of the core. Consider the Boolean variables A (alarm sounds), FA (alarm is faulty), and FG (gauge is faulty) and the multivalued nodes G (gauge reading) and T (actual core temperature).

Draw a Bayesian network for this domain, given that the gauge is more likely to fail when the core temperature gets too high.

⦁ (20pt) Two astronomers in different parts of the world make measurements M1 and M2 of the number of stars N in some small region of the sky, using their telescopes. Normally, there is a small possibility e of error by up to one star in each direction. Each telescope can also be badly out of focus , in which case the scientist will undercount by three or more stars (or if N is less than 3, fail to detect any stars at all). Consider the three networks shown.

⦁ Which of these Bayesian networks are correct representations of the preceding information?
⦁ Which is the best network? Explain.

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