Question 3 [25 points]: One method for approximate inference in Bayesian Networks is the Markov Chain Monte Carlo (MCMC) approach. This method depends on the important property that a variable in a Bayesian network is independent from any other variable in the network given its Markov Blanket. P(C) = .5 U Um Cloudy C P(S) C P(R) 10 Sprinkler Rain X 80 211 50 20 Wet Grass S R P(W) Yn .99 T F 90 T 90 .00 Figure 3: (left) The Markov Blanket of variable X (right) The Rain/Sprinkler network. a) Prove that P(X | MB(X)) = a P(X | U1,...,Um)Ily: P(Yi | Zil ...) where MB(X) is the Markov Blanket of variable X. b) Consider the query P(Rain Spronkler = true, WetGrass = true) in the Rain/Sprinkler network and how MCMC would answer it. How many possible states are there for the approach to consider given the network and the available evidence variables? c) Calculate the transition matrix Q that stores the probabilities P(y->y') for all the states y,y'. If the Markov Chain has n states, then the transition matrix has size nxn and you should compute n' probabilities