Let us express jd as 1 d suppose that although

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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 Metropolis-Hastings 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 Metropolis-Hastings 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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