1 Consider the query P(Rain | Sprinkler = true,WetGrass = true) in Figure 12

(a) and how Gibbs sampling can answer it.

a. How many states does the Markov chain have?

b. Calculate the transition matrix Q containing q(y → y′) for all y, y′.

c. What does Q2, the square of the transition matrix, represent?

d. What about Qn as n → ∞?

e. Explain how to do probabilistic inference in Bayesian networks, assuming that Qn is available. Is this a practical way to do inference?