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Bayesian Additive Distribution Regression

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Nonparametric and PAC-Bayes Bayesian learning · 0 citations (OpenAlex)

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Proposes DistBART, a BART-prior model over the Riesz representer of a linear functional for distribution regression, with kernel connections, a posterior contraction result, and a scalable random-tree-feature approximation.

Problem

Predicting a scalar response from a distribution-valued (grouped sample) predictor, where existing kernel-mean-embedding and aggregation methods ignore sparse additive structure and rely on hand-chosen summaries.

Approach

Model the regression functional as a linear functional int psi(x) G(dx) with psi given a BART prior, so that shallow trees induce sparse additive decompositions over low-dimensional marginals of G; this equals GP/kernel ridge regression with a data-adaptive kernel, and a random-feature horseshoe approximation makes it scalable.

Key results

• On the voting data, the non-linear DistBART variant achieved the best performance, followed by linear RBF-embedding and linear DistBART, while mean features performed much worse.
• In synthetic experiments with exponential marginals, BART features gave a sizable RMSE improvement over RBF features; with multivariate normal G_i, RBF features performed well.
• Theory: the DistBART fractional posterior contracts at rate epsilon_N, with BART prior thickness giving near-minimax (log N / N)^(alpha/(2alpha+d)) for (d,S)-sparse additive functions.

Surprising findings

• With multivariate normal group distributions, RBF kernel features outperformed BART features, contrary to the sparse-additive motivation.
• Additive summary of the voting model had only about 30% summary-R^2 for age, suggesting strong interactions that additive summaries miss.

Limitations

• Assumes additive decomposition over low-dimensional marginals, which may fail for image-like or spatially structured inputs.
• Measurement error not accounted for; relevant when M_i is small relative to N.
• Posterior contraction theory requires boundedness assumption and inner sample size conditions.

Experimental setup

Datasets: 2016 US presidential election PUMA data from American Community Survey synthetic Gaussian-copula data

Benchmarks: synthetic distribution regression simulations 2016 US presidential election voting data

Metrics: RMSE R^2 (squared correlation on test data) LOCO variable importance summary-R^2

Baselines: RBF kernel mean embedding features mean features sliced Wasserstein kernel ridge regression lasso soft BART

Builds on: BART kernel mean embeddings Safe Bayes horseshoe prior

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