The Metropolis€“Hastings algorithm is a member of the MCMC family; as such, it is designed to generate samples x (eventually) according to target probabilities Ï€(x). (Typically we are interested in sampling from Ï€(x)=P(x | e).) Like simulated annealing, Metropolis€“Hastings operates in two stages. First, it samples a new state x€² from a proposal distribution q(x€² | x), given the current state x. Then, it probabilistically accepts or rejects x€² according to the acceptance probability

If the proposal is rejected, the state remains at x.

a. Consider an ordinary Gibbs sampling step for a specific variable X_i. Show that this step, considered as a proposal, is guaranteed to be accepted by Metropolis€“Hastings. (Hence, Gibbs sampling is a special case of Metropolis€“Hastings.)

b. Show that the two-step process above, viewed as a transition probability distribution, is in detailed balance with π.

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т (х)q(x|x) п (х)q(x'|x) |a (х'|x) — min (1, п