  We rigorously prove that deep Gaussian process priors can outperform Gaussian
process priors if the target function has a compositional structure. To this
end, we study information-theoretic lower bounds for posterior contraction
rates for Gaussian process regression in a continuous regression model. We show
that if the true function is a generalized additive function, then the
posterior based on any mean-zero Gaussian process can only recover the truth at
a rate that is strictly slower than the minimax rate by a factor that is
polynomially suboptimal in the sample size $n$.
