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 dx covariates, the rate over classes with degree smoothness s is n−s/(2s+dx+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.
Related
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BibTeX
@misc{norets2026adaptive,
title = {{Adaptive Bayesian Estimation of Conditional Densities Based on Orthogonal Polynomials}},
author = {Andriy Norets and Marco Stenborg Petterson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39177},
eprint = {2610.39177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39177},
}
RIS
TY - UNPB
TI - Adaptive Bayesian Estimation of Conditional Densities Based on Orthogonal Polynomials
AU - Andriy Norets
AU - Marco Stenborg Petterson
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39177
UR - https://arxiv.org/abs/2610.39177
AB - 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.
ER -