The paper addresses itself to Bayesian estimation of the function \[ F(x_ 1,x_ 2)=m+a(x_ 1)+b(x_ 2)+c(x_ 1,x_ 2) \] in the model \(y_ i=F(x_{1i},x_{2i})+e_ i\). A prior for F is constructed by putting independent priors on m,a,b, and c. They are normal distribution and Brownian motion. The proposed estimate of F is the limit of Bayesian estimates as Var(m)\(\to \infty.\)
It is demonstrated that the estimator corresponds to the minimum of \(\sum (y_ i-F(x_{1i},x_{2i}))^ 2+P(F)\), where P(F) is a suitable penalty. Asymptotic properties are considered also in the grid case, and relations to variance components analysis are examined.