We derive several selection criteria for generalized linear models following the principle of Minimum Description Length (cf. Rissanen [5], Barron et al, [2], and Hansen and Yu [4]). We focus our attention on the mixture form of MDL. Normal or normal-inverse gamma distributions are used to construct the mixtures, depending on whether or not we choose to account for possible over-dispersion in the data. In the latter case, we apply Efxon's [3] double exponential family characterization of GLMs. Standard Laplace approximations are then employed to derive computationally tractable selection rules. Each of the resulting criteria has an adaptive penalty on model complexity.
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