See the toy example following theorem 103 find the

Question

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.