Econ226_VIII

Econ226_VIII - 1 VIII. Model selection A. Marginal...

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Unformatted text preview: 1 VIII. Model selection A. Marginal likelihood Suppose were trying to choose among a series of models: Model 1: p y | 2 1 B Model M : p y | 2 M where 2 m are possibly of different dimension The Bayesian might think in terms of an unobserved random variable: s 1 if Model 1 is true B s M if Model M is true 2 and assign prior probabilities = 1 Pr s 1 B = M Pr s M with associated priors on the parameters p 2 1 | s 1 B p 2 M | s M From such a perspective, the probability that Model m is true given the data is p s m | y = m ; p y | 2 m p 2 m | s m d 2 m ! j 1 M = j ; p y | 2 j p 2 j | s j d 2 j q = m p m y ! j 1 M = j p j y The expression p m y ; p y | 2 m p 2 m | s m d 2 m is sometimes called the "marginal likelihood" of Model m 3 The Bayesian would say that the data favor the model for which p s m | y is biggest. With diffuse priors = m 1/ M this is equivalent to choosing the model with the highest marginal likelihood. VIII. Model selection A. Marginal likelihood B. Schwarz criterion First lets examine the behavior of p m y ; p y | 2 m p 2 m | s m d 2 m as the sample size T gets large 4 Suppose log p y | 2 !...
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Econ226_VIII - 1 VIII. Model selection A. Marginal...

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