Econ226_VIII

# Log t for k the dimension of 2 choosing the model m

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log T for k the dimension of 2 Choosing the model m for which log p y | 2 ± mT   " k m /2   log T is biggest is known as the using the Schwarz Information Criterion (SIC) or Bayesian Information Criterion (BIC)

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7 Since it is asymptotically a Bayesian decision rule, SIC inherits the properties of being asymptotically admissible and consistent However, note that this result required the same regularity conditions needed to get asymptotic Normality of MLE VIII. Model selection A. Marginal likelihood B. Schwarz criterion C. Calculating the marginal likelihood with the Gibbs sampler
8 Goal: calculate p y   ± ; p y | 2   p 2   d 2 Couldn’t we get this by drawing 2 j   j ± 1,..., J from p 2   and then p ³ y   ± J " 1 ! j ± 1 J p y | 2 j     ? Answer: no, this algorithm is badly behaved numerically. Chib’s idea: think of evaluating at a point with a lot of mass (say the posterior mean 2 ' ). Note that for any 2 ' we have the identity p 2 ' | y   p y   ± p y | 2 '   p 2 '   p y   ± p y | 2 '   p 2 '   / p 2 ' | y   In many applications, we know p y | 2 '   and p 2 '   analytically (evaluating the likelihood and prior at posterior mean, respectively), but couldn’t calculate p 2 ' | y   explicitly

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9 Suppose we’ve generated draws from a two-block Gibbs sampler: p 2 1 | 2 2 , y   and p 2 2 | 2 1 , y   The object of interest is given by p 2 1 ' , 2 2 ' | y   ± p 2 1 ' | 2 2 ' , y   p 2 2 ' | y   where we may know p 2 1 ' | 2 2 ' , y   analytically.
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• Winter '09
• JamesHamilton
• Gibbs, Akaike information criterion, Bayesian information criterion, y|, Deviance information criterion

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