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See the toy example following Theorem 10.3. Find the transition matrix for the Markov chain constructed by the Metropolis-Hastings algorithm. Show that π = (0.1, 0.2, 0.3, 0. 4) is the stationary distribution and that the detailed balance condition is satisfied.
Theorem 10.3
(MCMC: Metropolis-Hastings algorithm). Let X0 be an arbitrary initial state. The sequence of random variables X0, X1, X2,... constructed by the aforementioned algorithm is a time-reversible Markov chain whose limiting distribution is π.
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