# ³ 1 y 2 2 j 2 3 j 7 under quite general conditions

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³ 1   | Y , 2 2 j   , 2 3 j

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7 Under quite general conditions, the realizations from a Markov chain for D v . converge to draws from the ergodic distribution of the chain = 2   satisfying = 2 j ³ 1     ± ; g k = 2 j ³ 1   | 2 j     = 2 j     d 2 j   Claim: the ergodic distribution of this chain corresponds to the posterior distribution: = 2   ± p 2 | Y   Proof: ; g k = 2 j ³ 1   | 2 j     = 2 j     d 2 j   ± ; g k p 2 3 j ³ 1   | Y , 2 1 j ³ 1   , 2 2 j ³ 1     p 2 2 j ³ 1   | Y , 2 1 j ³ 1   , 2 3 j     p 2 1 j ³ 1   | Y , 2 2 j   , 2 3 j     p 2 j   | Y   d 2 j
8 ± ; g k p 2 j ³ 1   , 2 j   | Y   d 2 j   ± p 2 j ³ 1   | Y   Implication: if we throw out the first D 0 draws (for D 0 large), then 2 D 0 ³ 1   , 2 D 0 ³ 2   , ..., 2 D   represent draws from the posterior distribution p 2 | Y   . Checks: (1) Change 2 1   ´ same answer? (2) Change D 0 , D ´ same answer? (3) Plot 2 j   as function of j .

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9 I. Bayesian econometrics F. Numerical Bayesian methods 1. Importance sampling 2. The Gibbs sampler 3. Metropolis-Hastings algorithm Suppose £ s t ¤ t ± 1 T is an ergodic K -state Markov chain, s t ± £ 1,2,..., K ¤ with transition probabilities p ij ± Pr ¡ s t ± j | s t " 1 ± i ¢ ! j ± 1 K p ij ± 1 for i ± 1,..., K p ij u 0 for i , j ± 1,..., K
10 The ergodic or unconditional probabilities satisfy Pr ¡ s t ± j ¢ ± ! i ± 1 K Pr ¡ s t ± j , s t " 1 ± i ¢ = j ± ! i ± 1 K p ij = i Proposition: Suppose we can find a set of numbers f 1 , f 2 ,.., f K such that f j u 0 for j ± 1,..., K !

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• Winter '09
• JamesHamilton
• Econometrics, Markov chain, Markov chain Monte Carlo

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