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- For numerical implementation of the BMA technique, some approximations are commonly found in the literature. The Laplace approximation described by Volinsky et al (1997) is adopted in this paper. This approximation is shown in the following equation log( (DjMm)) l dk log(n) (21) where dk represents the number of parameters to estimate and l denotes the loglikelihood evaluated in the estimated parameters. (21) is called the Bayesian information criterion (BIC) approximation showed by Hoeting (1999).
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- Mm2M ( j D; Mm) (D j Mm) (Mm) where (D j Mm) is the likelihood of the data given the particular model Mm 2 M, and (Mm) represents the prior density de…ned over M. Basically these results show that the posterior density of the parameter is a weighted average of the conditional densities of the parameter for dierent assumptions about the true model. This technique is in general called Bayesian Model Averaging. (BMA) Leamer (1978) provides expressions for the conditional expectation and variance of given the set of data D: E( j D) = X Mm2M (Mm j D)E( j D; Mm) and var( j D) = E( 2 j D) (E( j D))2 = X Mm2M (Mm j D)var( j D; Mm) + X Mm2M (Mm j D)(E( j D; Mm) E( j D))2 (20) where (Mm j D) = (D j Mm) (Mm) P Mm2M (D j Mm) (Mm) therefore, the conditional variance of given the set of data D in (20) is broken down into two additive components: an intra-model variance and an across-models variance.
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