Adaptive Bayesian Estimation of Conditional Densities Based on Orthogonal Polynomials

Abstract

We study Bayesian estimation of conditional densities represented by normalized squares of orthogonal polynomial series. We specify the conditional density without modeling the marginal distribution of covariates. A prior on smoothness, the number of terms, and the coefficients gives adaptive posterior contraction at the minimax rate without a logarithmic factor when the normalized square-root representation of the true density is bounded away from zero and sufficiently smooth. For one outcome and $d_x$ covariates, the rate over classes with degree smoothness $s$ is $n^{-s/(2s+d_x+1)}$. The posterior predictive density attains the corresponding minimax squared Hellinger risk. For bounded nonnegative representations in Legendre and Chebyshev classes, we establish the minimax rate at every $s>0$. A prior that also permits a small mixture with the reference density adapts over these classes up to a logarithmic factor and retains the exact rate on the positive, sufficiently smooth subclass. Additional results concern identification, conditional distribution functions and quantiles, coefficient recovery and uniform convergence, independent Gaussian coefficient priors, and selection of polynomial terms.