3. For Markov chain, answer the following.
a. Why is it that irrespective of where the chain begins, we can
be sure that it will eventually end up sampling from the posterior distribution if we run it long enough? b. Write down the formula for acceptance kernel α. Explain how do you use α to decide if the value θ' sampled from the candidate density is accepted or rejected? Where does the Markov chain move to if θ' is accepted? And when θ' is not accepted?
c. Explain the rationale behind why α is formulated the way it is? Explain the logic behind whether a particular move is accepted or rejected? d. Show what happens to α if we choose a symmetric, random walk candidate density?
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