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Unformatted text preview: ral is the key to Monte Carlo methods.
Step 4. With probability α(θt−1 , θ∗ ), accept the move θt−1 → θ∗ by setting θt =
θ∗ and incrementing t ← t + 1. Otherwise, discard θ∗ and stay at θt−1 .
Step 5. Until stationary distribution and the desired number of draws are reached,
return to Step 2.
Equation (25) reduces to what we developed intuition for in (24) when the
proposal distribution is symmetric: J (θ, θ' ) = J (θ' , θ). We will see in the
next theorem that the extra factors in (25) are necessary for the posterior to
be the stationary distribution. 26 6.2.3 Convergence of MetropolisHastings Because the proposal distribution and α(θt−1 , θ∗ ) depend only on the cur
rent state, the sequence θ0 , θ1 , . . . forms a Markov chain. What makes the
MetropolisHastings algorithm special is the following theorem, which shows
that if we simulate the chain long enough, we will simulate draws from the
posterior.
Theorem 5. If J (θ, θ' ) is such that that the Markov chain θ...
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 Spring '12
 CynthiaRudin

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