FULLY BAYES FACTORS WITH A GENERALIZED g-PRIOR

where α is an unknown intercept parameter, 1n is an n× 1 vector each component of which is one, XF = (x1, . . . ,xp) is an n×p design matrix, βF is a p×1 vector of unknown regression coefficients, In is an n × n identity matrix and σ 2 is an unknown positive scalar. (The subscript F denotes the full model.) We assume that the columns of XF have been standardized so that for 1 ≤ i ≤ p, x′i1n = 0 and x′ixi/n= 1. We shall be particularly interested in the variable selection problem where we would like to select an unknown subset of the important predictors. It will be convenient throughout to index each of these 2 possible subset choices by the vector γ = (γ1, . . . , γp)′, where γi = 0 or 1. We use qγ = γ ′1p to denote the size of the γ th subset. The problem then becomes that of selecting a submodel of (1.1) which has a density of the form p(y|α,βγ , σ 2,γ )= φn(y;α1n + Xγβγ , σ In), (1.2)