Question: Markov chain sampler algorithm Let 7r = (71'1,7r2, ...) be a probability mass function on a countable state-space S. An MCMC method known as Barker's

Markov chain sampler algorithm

Let 7r = (71'1,7r2, ...) be a probability mass function on a countable state-space S. An MCMC method known as Barker's algorithm constructs a Markov chain on S, which given its current state 2' proceeds by randomly drawing another state 3' with proposal probability qz-j and then accepting this proposed state with probability \"Mi 0).. = 7993's + Wiqz'j U It will stay at the current state if the proposal is not accepted. 1. Find the transition probabilities pij of Barker's Markov chain. 2. Show that at is a stationary distribution of Barker's Markov chain. Is the chain reversible? 3. Compare the behavior of Barker's algorithm and the Metropolis-Hastings algorithm. In doing so, consider the case of symmetric proposals ('L'j = q) and think about the cases where current state i and proposed state j have 71'2- > 7rj. Which method seems preferable

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