On a sub-optimality conjecture

We consider the problem of estimating the response function in a random design regression model with Gaussian errors. We confine ourselves to regression functions with a known degree of smoothness. We investigate the asymptotic behaviour of the risk (using integrated squared error loss) of the posterior mean resulting from independent normal priors on the coefficients in a series expansion of the regression function through an orthogonal basis. We show that the Bayes’ estimator corresponding to any product Gaussian prior supported on the parameter space involved cannot attain the optimal minimax rate over ellipsoids containing the true value of the parameter. This result provides support for a recently posed “conjecture” according to which, in the present nonparametric regression problem, product Gaussian priors having a special power-variance structure, restricted to ellipsoids, lead to posterior distributions with sub-optimal rates.