1 The operation of arc reversal in a Bayesian network allows us to change the direction of an arc X → Y while preserving the joint probability distribution that the network represents

(Shachter, 1986). Arc reversal may require introducing new arcs: all the parents of X also become parents of Y , and all parents of Y also become parents of X.

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a. Assume that X and Y start with m and n parents, respectively, and that all variables have k values. By calculating the change in size for the CPTs of X and Y, show that the total number of parameters in the network cannot decrease during arc reversal. (Hint: the parents of X and Y need not be disjoint.) b. Under what circumstances can the total number remain constant? c. Let the parents of X be UUV and the parents of Y be VU W, where U and W are disjoint. The formulas for the new CPTs after arc reversal are as follows: P(Y | U, V, W) = == P(Y | V, W, x)P(x | U, V) I P(X | U, V, W,Y) = P(Y |X, V, W)P(X |U, V)/P(Y | U, V, W). Prove that the new network expresses the same joint distribution over all variables as the original network.