where ρ is the posterior mean of ρ, Σ2,1,1 is the first diagonal element of Σ2 , k 2 (x * ) = [κ 2 (x * , x 21 ; φ 2 ), . . . , κ 2 (x * , x 2n 2 ; φ 2 )] T is the vector of correlations between the unobserved structure x * and the training set for the highest level response and fh (x * ) = {E[y l (x * )|y l ], 1}. Sampling from the posterior predictive distribution can proceed via Monte Carlo methods or by using a tdistribution with 2α 2 + n 2 degrees of freedom, mean (4) and vari-ance (5). After checking the adequacy of this approximation for our examples, we adopt this latter approach for computational convenience.