1.(20 pts) (Exercise 14.21) Consider the query P(Rain | Sprinkler = true, WetGrass = true) in the figure below and how Gibbs sampling can answer it. Sprinkler CP(Sc) .10 50 P(C=.5) Cloudy WetGrass SRP(Ws.r) 99 .90 tt ft .90 .00 Rain T f P(RC) 80 20 (a) (2 pts) How many states does the Markov chain for this query have? (b) (10 pts) Calculate the transition matrix Q containing the kernel k(y y') for every directed edge from, say, a state y to another state y' or itself (in the case y = y'), in the Markov chain. Label the rows and columns of the matrix by the states such that an entry in the matrix corresponds to the probability of a transition from the state given by the row label to the state given by the column label. (c) (2 pts) What does Q, the square of the transition matrix, represent? (d) (3 pts) What about Q" as n ? (e) (3 pts) Explain how to do probabilistic inference in a Bayesian network, assuming that Q" is available. Is this a practical way to do inference?