1 Motivation We are interested in Bayesian inference and prediction with feed-forward neural network models (FFNN's), speciically those with one hidden layer with M hidden nodes, p input nodes, 1 output node and logistic activation functions: we try to predict a variable y in terms of p variables x = (x 1 ; :::; x p), with regression function y(x) = P M i=1 j (j x) where (z) = exp(z) 1+exp(z). ...