SBAMDT: Bayesian Additive Decision Trees with Adaptive Soft Semi-multivariate Split Rules

Bledar A. Konomi; Huiyan Sang; Ligang Lu; Stamatina Lamprinakou

REVIEW 4 major objections 5 minor 42 references

SBAMDT: Bayesian Additive Decision Trees with Adaptive Soft Semi-multivariate Split Rules

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read SBAMDT's node-level mixture of hard and soft splits beats BART, SBART, and BAMDT on spatial regression benchmarks.

desk verdict A promising BART extension for spatial data whose empirical claims overreach, but the core idea deserves a close look. read the letter →

arxiv 2501.09900 v1 pith:7A4EWFIP submitted 2025-01-17 stat.ML cs.LGmath.STstat.MEstat.TH

Stamatina Lamprinakou , Huiyan Sang , Bledar A. Konomi , Ligang Lu This is my paper · ORCID

classification stat.MLcs.LGmath.STstat.MEstat.TH MSC 62G0862F1562M30

keywords Bayesianadditiveregressiontreessoftdecisionrulessemi-multivariatesplitsspatialnonparametricgraphLaplacianembeddingminimumspanningtreebipartitionGaussianprocesscovarianceMCMCbackfitting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes SBAMDT, a Bayesian additive tree ensemble for nonparametric regression in which every internal node chooses between a hard split and a soft, probabilistic split, and in which soft splits can use multivariate structured features as well as single unstructured features. Its aim is to show that this adaptive hard-soft rule lets one ensemble represent sharp discontinuities and smooth transitions simultaneously, and it reports that SBAMDT consistently outperforms the BART, SBART, and BAMDT baselines on U-shaped, square-domain, and New York City education data. A careful reader would care because spatial regression on complex domains—road networks, city boundaries, brain surfaces—has typically forced a choice between axis-aligned trees that ignore geometry and rigid hard multivariate splits that cannot capture smooth variation; SBAMDT is offered as a single tool that does both. The claim is carried by a probabilistic leaf-assignment rule based on distances to nearest reference knots, with multivariate split geometry supplied by a graph-Laplacian embedding and spanning-tree bipartitions.

What carries the argument

The central object is the adaptive soft semi-multivariate split rule at each internal node $\eta$. It is a categorical choice $A_\eta \in \{0,1,\dots,k\}$: $A_\eta=0$ is a hard split that sends a point $d$ to the nearer child with probability one, while $A_\eta=c$ is a logistic gate $z_{\eta L}(d)=\left(1+\exp(-\alpha_c(d_{\eta R}(d)-d_{\eta L}(d))/C_\eta)\right)^{-1}$, where $d_{\eta L}(d)$ and $d_{\eta R}(d)$ are the distances from $d$ to the nearest reference knots in the left and right children and $C_\eta$ is a normalizing constant. Univariate splits use a single unstructured feature; multivariate splits first embed the structured-feature reference knots through the normalized graph Laplacian and then remove an edge from a minimum spanning tree, so the split boundary follows the geometry of the domain. The node probabilities multiply along a root-to-leaf path into leaf-assignment probabilities $\phi_{hl}(d)$, and the ensemble $f(d)=\sum_{h=1}^{m}\sum_{l=1}^{L_h}\mu_{hl}\phi_{hl}(d)$ with Gaussian leaf weights is the model being fit. Theorem 3.1 identifies the induced covariance $C(f_i,f_j)=\frac{\beta_\mu}{\alpha_\mu-1}\sum_{h=1}^{m}\sum_{l=1}^{L_h}\phi_{hl}(d_i)\phi_{hl}(d_j)$, which turns these probabilities into coherent uncertainty quantification.

What would settle it

On a noiseless square-domain regression surface that is exactly piecewise constant with axis-aligned jumps, run BART and Sk-BAMDT with the paper's default settings but a larger tree ensemble; if BART's RMSPE or CRPS falls below Sk-BAMDT's, the claim that SBAMDT consistently outperforms BART is false. The paper's square example already edges toward this regime but includes smooth pieces, so the sharp piecewise-constant version is the decisive check.

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Extended reading notes

Core claim

The central claim, stated in the paper's conclusion, is that SBAMDT consistently outperforms BART, SBART, and BAMDT for nonparametric regression with structured and unstructured features. In the U-shape simulation, the Sk-BAMDT variant attains the lowest root mean square prediction error, mean absolute prediction error, and continuous ranked probability score of the four methods; in the square-domain example, which was designed to favor axis-parallel BART, Sk-BAMDT still wins; and on the NYC education data both SBAMDT variants produce the lowest metrics while following spatial boundaries more closely than BART and SBART and adapting to smooth regions that BAMDT's hard splits cannot. The paper attributes these wins to the mixture of hard and soft decisions at each node rather than soft splits everywhere (SBART) or hard splits everywhere (BAMDT). It also proves in Theorem 3.1 that, conditional on trees and decision types, the prior and posterior of the fitted function are valid Gaussian processes, so predictive intervals come from a coherent stochastic-process model rather than an ad hoc covariance.

Load-bearing premise

The method's central gamble is that replacing true distances along the surface with ordinary distances in a graph-based coordinate embedding does not distort the geometry that determines the splits, and that the soft-split formulas stay smooth where the nearest reference knot changes; both assumptions are untested in the paper.

Editorial extensions

If this is right

If the reported comparisons hold, SBAMDT becomes a better default than BAMDT for spatial regression on complex domains, because multivariate soft splits capture smooth within-cluster variation while hard splits preserve sharp boundaries.
The ensemble can represent functions smoother than the piecewise-constant surfaces of ordinary BART, since the logistic gates contribute smooth basis functions; this lifts a practical ceiling on hard-tree ensembles.
Posterior draws of the decision indicators $A_\eta$ provide a per-node description of whether a learned boundary is sharp or gradual, so the model output also says where the fitted function changes suddenly.
Because the fitted function is a valid Gaussian process conditional on trees and decision types, users get predictive intervals without having to specify a covariance kernel.
Replacing geodesic $k$-nearest-neighbor construction with a graph-Laplacian embedding reduces the computational cost of MST-based multivariate splits, which matters for scaling to larger spatial datasets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implication the paper leaves implicit is that the covariance in Theorem 3.1 is a natural, data-driven kernel for kriging-style interpolation or Bayesian optimization on manifolds, since the same leaf-assignment products define similarity without a hand-chosen metric.
The hard/soft indicators could be summarized across MCMC iterations as a posterior 'discontinuity map' of the fitted surface, giving applied users a spatially explicit measure of where transitions are abrupt; the paper computes importance metrics but not this summary.
A testable extension is to put a hyperprior on the Laplacian embedding dimension $k$, which the paper never specifies or tunes, or to compare the embedded distances against true geodesic distances on the structured domain before fitting; this would show whether the multivariate split quality depends on that silent choice.
A controlled experiment with all splits univariate, and structured features excluded, would isolate whether the hard-soft mixture alone reproduces SBART's smoothness adaptation; the paper's comparisons always involve structured multivariate splits.

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Formalized claims in Lean

Claim #1: The central claim, stated in the paper's conclusion, is that SBAMDT consistently outperforms BART, SBART, and BAMDT for nonparametric regression with structured and unstructured features. In the U-shape simulation, the Sk-BAMDT variant attains the lowest root mean square prediction error, mean absolute prediction error, and continuous ranked probability score of the four methods; in the square-dom

/-- @claim 1 The central claim, stated in the paper's conclusion, is that SBAMDT consistently outperforms BART, SBART, and BAMDT for nonparametric regression with structured and unstructured features. In the U-shape simulation, the Sk-BAMDT variant attains the lowest root mean square prediction error, mean absolute prediction error, and continuous ranked probability score of the four methods; in the square-dom -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read SBAMDT. The core idea is genuinely new and useful: replacing BAMDT's hard multivariate splits with a node-specific mixture of hard and soft logistic gates based on distances to nearest reference knots, and using a graph-Laplacian embedding to approximate manifold distances. That is a reasonable upgrade for spatial problems with complex domains, and the Sk-BAMDT variant does well in the reported experiments. The MCMC details are spelled out more thoroughly than in many BART papers, which is a plus.

The problem is the conclusion. "Consistently outperforms" is not supported by their own tables. S2-BAMDT, the variant closest to SBART, loses to BART on the square domain (RMSPE 11.35 vs 10.76), and its MAPE and CRPS edges there are modest. Sk-BAMDT leads, but its smoothness multiplier q is tuned per dataset via cross-validation, and the baselines are not given equivalent tuning. That asymmetry alone could explain part of the reported gain. The authors should report a fairer comparison: tune all methods on the same criterion, or fix defaults for all.

The GP-connection theorem (3.1) is largely a repackaging of the known BART-as-GP result, with a proof gap that matters. Appendix L differentiates the logistic soft split as though the nearest knots were fixed, but those knots change at Voronoi boundaries. So the mean-square differentiability claim does not follow from the displayed expression. This is fixable by stating the derivative holds on open cells and addressing the kinks separately, or by proving differentiability of the composition more carefully.

No code or data is released, which limits reproducibility, especially since the empirical claims rest on a handful of simulations.

I would send this to peer review: the core method is a genuine contribution, and the issues are all addressable in revision. The paper is for spatial statisticians and Bayesian nonparametrics researchers interested in tree-based models that handle soft transitions and complex geometry. Not yet a practical default, but a solid starting point.

Referee Report

4 major / 5 minor

Summary. The paper proposes SBAMDT, an additive Bayesian decision-tree regression model that combines hard and soft splits at each internal node, uses graph-Laplacian embeddings of structured multivariate features for multivariate splits based on minimum spanning trees, and assigns observations to children via distances to reference knots. Two variants are introduced: Sk-BAMDT, with a small discrete set of smoothness levels, and S2-BAMDT, with a tree-specific smoothness parameter. The model is evaluated on two synthetic examples and a New York City education dataset against BART, SBART, and BAMDT, with a claimed consistent improvement; a Gaussian-process connection is stated in Theorem 3.1 and proved in Appendix L.

Significance. The modeling idea is useful and the paper provides substantial implementation detail: full MCMC proposals, conditional distributions, and appendices for both variants. The GP connection is a nice theoretical observation, though the differentiability part is not fully established. If the empirical claim were supported, SBAMDT would offer a practical default for spatial nonparametric regression with complex domains. However, the current evidence base contains a contradictory result and a tuning asymmetry that prevent the strong 'consistently outperforms' claim from being accepted as stated.

major comments (4)

[Table 2 and Section 5.2] Table 2 shows S2-BAMDT with RMSPE 11.35 (sd 3.46) on the square example, worse than BART at 10.76 (1.39) and BAMDT at 10.34 (1.06); yet Section 7's concluding sentence asserts that 'SBAMDT consistently outperforms BART, SBART, and BAMDT'. The conclusion therefore overstates what the experiments show: at best only Sk-BAMDT supports the claim, and only on the U-shape and NYC examples. Please restrict the claim to Sk-BAMDT or provide an explanation for the discrepancy and show that S2-BAMDT's underperformance does not undermine the central message. Additionally, the Sk-BAMDT RMSPE in Table 2 has a standard deviation of 3.76, which is large relative to the difference from BART, so the reported advantage for the winning variant may not be stable.
[Section 3.3.0.2 and Tables 1-3] The smoothness multiplier q is tuned per dataset for the SBAMDT variants (q=8 for U-shape, q=8/10 for square, q=10/6 for NYC, and q=6/4 in Appendix K) using cross-validation/WAIC, but no equivalent hyperparameter tuning is reported for BART, SBART, or BAMDT, which are presumably run with default settings. Under this asymmetry, part of the reported gain may be attributable to tuning effort rather than to the adaptive soft-split mechanism. Please add tuned baselines or a sensitivity analysis showing the results are robust to baseline hyperparameters.
[Appendix L, Eq. (8)] The derivative of the soft multivariate split probability in Eq. (8) is computed as if the nearest left and right reference knots s_L* and s_R* are fixed, but these nearest knots change at Voronoi boundaries; at those boundaries the expression is not the derivative and the soft assignment has a kink. Consequently, the mean-square differentiability assertion in Theorem 3.1(1), which relies on this expression in 'the absence of hard splits', is not established. Please either prove differentiability across the Voronoi boundaries or weaken the statement to differentiability on the open cells where the nearest knots are constant.
[Section 3.1] The graph-Laplacian embedding uses the k eigenvectors corresponding to the k smallest non-zero eigenvalues, but the value of k is never specified or tuned in any of the experiments (U-shape, square, NYC, Appendix K). Since the multivariate split geometry and the faithfulness of Euclidean distance in the embedded space depend on k, the experimental protocol is not fully reproducible and the manifold-embedding premise is untested. Please report the chosen k for each dataset and provide a sensitivity check.

minor comments (5)

[Section 4] The labels 'Approach 1' and 'Approach 2' are swapped relative to their definitions in Section 3.2: in Section 4, Approach 1 (Multinomial/Dirichlet) corresponds to Sk-BAMDT, while Section 3.2 calls this Approach 2; similarly, Approach 2 in Section 4 (Gamma/Bernoulli) is Section 3.2's Approach 1 (S2-BAMDT). This will confuse readers trying to map the algorithm to the model variants.
[Section 3.1] In the univariate split description, 'x∗ j(η) is the jth structured feature' should read 'unstructured feature', since the univariate split uses x.
[Section 5.2] The text says 'we consider a noise-free environment' but then generates data with σ=0.1; please reconcile this inconsistency.
[Figure numbering] The text at the start of Section 3.2 says 'Figure 1 compares hard and soft decision trees', but Figure 1 earlier shows the spanning tree bipartition; the comparison figure is actually Figure 2.
[Section 3.3.0.2] 'Watanable-Akaike information criterion' is a typo; also the Gelman et al. (2014) citation is repeated for both WAIC and cross-validation, but the latter is standard and could use an explicit reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SBAMDT's soft-split model and its evaluation are self-contained; the GP theorem is a direct consequence of model assumptions, and the cited BAMDT building blocks are externally published.

full rationale

The paper's central product is an estimator rather than a quantity derived from its own inputs. The soft-split probabilities, hard/soft decision indicators, and leaf weights are defined directly by the model hierarchy in Sections 3.2 and 4, and prediction is evaluated on held-out test data against external baselines (BART, SBART, BAMDT). No parameter is fitted to a subset and then presented as a prediction of that same subset. The per-dataset tuning of q for Sk-BAMDT via cross-validation/WAIC is standard model selection and does not make the reported RMSPE/MAPE/CRPS reductions 'forced by construction.' The GP connection in Theorem 3.1 is an immediate consequence of zero-mean normal leaf priors and independent tree summation; it is a mathematical characterization, not an assumption used to fit the model or to justify the empirical wins. The overlapping-author citation Luo et al. (2022) provides the BAMDT multivariate partition machinery, but that work is a published, externally checkable baseline and is not invoked as a uniqueness theorem or as the source of the soft-split contribution. There are legitimate correctness risks—the differentiability proof in Appendix L differentiates through nearest-knot assignments without addressing Voronoi-boundary kinks, and the 'consistently outperforms' conclusion is somewhat undercut by tuning asymmetry and by S2-BAMDT's square-domain loss—but these are evidentiary concerns, not circular reductions. No equation or claim in the paper reduces by definition to its own inputs.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The method relies on several hand-chosen values and modeling premises. Most importantly, the smoothness scale q is tuned per dataset, the Sk-BAMDT alpha grid is hand-set, and the graph-Laplacian embedding dimension k is unspecified. The axioms include the manifold approximation assumption and the validity of the logistic distance-based soft assignment.

free parameters (3)

q = q=8,10,6,4 across scenarios
Smoothness scaling tuned per dataset using WAIC and cross-validation (Section 3.3.0.2); directly controls the sharpness of soft splits and affects all performance metrics.
alpha grid = {0.5, 1, 2} times q
Hand-chosen discrete set of softness levels for Sk-BAMDT (Section 3.3.0.2); the model selects among these categories at each node.
Laplacian embedding dimension k = not stated
The number of eigenvectors used in the graph Laplacian embedding (Section 3.1) is never specified or tuned; it determines the embedded coordinates for multivariate splits.

assumptions (3)

ad hoc to paper The graph Laplacian embedding with Euclidean distance approximates geodesic distances on the structured feature manifold.
Invoked in Section 3.1 to replace geodesic k-NN MST with an embedded-space MST; if false, multivariate splits do not respect the domain geometry.
domain assumption The logistic soft-split probability based on differences of distances to nearest left/right knots is a valid smooth transition model and can represent the regression function's smoothness.
The model definition in Section 3.2 relies on this; the basis functions have kinks where the nearest knot changes, which also affects the GP differentiability claim in Theorem 3.1.
standard math Tree components, node decisions, and leaf parameters are a priori independent (Galton-Watson tree prior, normal leaf priors, independent decision types).
Assumed in Section 3.3 following Chipman et al. (2010); standard in BART and used for the product-form likelihood and MCMC.

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Cite this review

Pith. "Pith review of SBAMDT: Bayesian Additive Decision Trees with Adaptive Soft Semi-multivariate Split Rules." pith.science (2026). https://pith.science/paper/7A4EWFIP

@misc{pith2026250109900,
  author       = {Pith},
  title        = {Pith review of: SBAMDT: Bayesian Additive Decision Trees with Adaptive Soft Semi-multivariate Split Rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7A4EWFIP}},
  note         = {Machine review of arXiv:2501.09900}
}

read the original abstract

Bayesian Additive Regression Trees [BART, Chipman et al., 2010] have gained significant popularity due to their remarkable predictive performance and ability to quantify uncertainty. However, standard decision tree models rely on recursive data splits at each decision node, using deterministic decision rules based on a single univariate feature. This approach limits their ability to effectively capture complex decision boundaries, particularly in scenarios involving multiple features, such as spatial domains, or when transitions are either sharp or smoothly varying. In this paper, we introduce a novel probabilistic additive decision tree model that employs a soft split rule. This method enables highly flexible splits that leverage both univariate and multivariate features, while also respecting the geometric properties of the feature domain. Notably, the probabilistic split rule adapts dynamically across decision nodes, allowing the model to account for varying levels of smoothness in the regression function. We demonstrate the utility of the proposed model through comparisons with existing tree-based models on synthetic datasets and a New York City education dataset.

Figures

Figures reproduced from arXiv: 2501.09900 by the authors.

**Figure 2.** Comparison of a hard decision tree to a soft decision tree as derived by Linero (SBART) and the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** The ground truth for f(s, x) and the predictive surfaces ˆf(s, x) of each method for a U-shape simulated data. The red circle indicates discontinuity boundaries in the true function projected to the 2-D U-shape domain. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: The APE of each method for one U-shape simulation. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: The CRPS of SBAMDT, BAMDT, BART and SBART associated with U-shape Example. [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: The ground truth for f(s, x) and the predictive surfaces ˆf(s, x) of SBAMDT, BAMDT, BART and SBART associated with Toy Example. grid points as potential univariate split cutoff values for each unstructured feature and executed MCMC algorithms for 20,000 iterations, ret…

**Figure 7.** Figure 7: The APE of SBAMDT, BAMDT, BART and SBART associated with Toy Example. [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: The observed and fitted value of income at observed locations for SBAMDT. [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]

**Figure 9.** Figure 9: The spatial partial dependence plots for SBAMDT, BAMDT, BART, and SBART. [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

**Figure 10.** Figure 10: The ground truth for f(s, x) and the spatial of SBAMDT, BAMDT, BART and SBART associated with U-shape Example 2. Red dashed circle indicate discontinuity surfaces in the true function projected to M. We use the product rule to find the gradient of the probability of a…

**Figure 11.** Figure 11: The APE of SBAMDT, BAMDT, BART and SBART associated with U-shape Example 2. [PITH_FULL_IMAGE:figures/full_fig_p031_11.png]

**Figure 12.** Figure 12: The CRPS of SBAMDT, BAMDT, BART and SBART associated with U-shape Example 2. [PITH_FULL_IMAGE:figures/full_fig_p032_12.png]

**Figure 13.** Figure 13: The sd of SBAMDT, BAMDT, BART and SBART associated with U-shape Example 2. [PITH_FULL_IMAGE:figures/full_fig_p033_13.png]

**Figure 14.** Figure 14: The importance metric for each feature given by Sk-BAMDT, S2-BAMDT, BAMDT, BART [PITH_FULL_IMAGE:figures/full_fig_p034_14.png]

**Figure 15.** Figure 15: The importance metric for each feature given by Sk-BAMDT, S2-BAMDT, BAMDT, BART [PITH_FULL_IMAGE:figures/full_fig_p036_15.png]

**Figure 16.** Figure 16: The predicted income versus the population over 25 with at least a bachelor’s degree. Colored [PITH_FULL_IMAGE:figures/full_fig_p037_16.png]

**Figure 17.** Figure 17: The importance metric for each feature for one U-shape simulation. [PITH_FULL_IMAGE:figures/full_fig_p038_17.png]

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Reference graph

Works this paper leans on

42 extracted references · 31 canonical work pages

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...
[2]

Bart: Bayesian additive regression trees

Hugh A Chipman, Edward I George, and Robert E McCulloch. Bart: Bayesian additive regression trees. 2010

2010
[3]

A decision-theoretic generalization of on-line learning and an application to boosting

Yoav Freund and Robert E Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of computer and system sciences, 55 0 (1): 0 119--139, 1997

1997
[4]

Bagging predictors

Leo Breiman. Bagging predictors. Machine learning, 24: 0 123--140, 1996

1996
[5]

Random forests

Leo Breiman. Random forests. Machine learning, 45: 0 5--32, 2001

2001
[6]

The B ayesian additive classification tree applied to credit risk modelling

Junni L Zhang and Wolfgang K H \"a rdle. The B ayesian additive classification tree applied to credit risk modelling. Computational Statistics & Data Analysis, 54 0 (5): 0 1197--1205, 2010

work page 2010
[7]

Multinomial probit B ayesian additive regression trees

Bereket P Kindo, Hao Wang, and Edsel A Pe \ n a. Multinomial probit B ayesian additive regression trees. Stat, 5 0 (1): 0 119--131, 2016

work page 2016
[8]

Variable selection for BART : an application to gene regulation

Justin Bleich, Adam Kapelner, Edward I George, Shane T Jensen, et al. Variable selection for BART : an application to gene regulation. T he A nnals of A pplied S tatistics , 8 0 (3): 0 1750--1781, 2014

work page 2014

Show all 42 references

[9]

Bayesian regression trees for high-dimensional prediction and variable selection

Antonio R Linero. Bayesian regression trees for high-dimensional prediction and variable selection. J ournal of the A merican S tatistical A ssociation , 113 0 (522): 0 626--636, 2018

2018
[10]

m BART : M ultidimensional M onotone BART

Hugh A Chipman, Edward I George, Robert E McCulloch, and Thomas S Shively. m BART : M ultidimensional M onotone BART . Bayesian Analysis, 1 0 (1): 0 1--30, 2021

2021
[11]

Bayesian nonparametric modeling for causal inference

Jennifer L Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20 0 (1): 0 217--240, 2011

2011
[12]

Nonparametric survival analysis using B ayesian additive regression trees ( BART )

Rodney A Sparapani, Brent R Logan, Robert E McCulloch, and Purushottam W Laud. Nonparametric survival analysis using B ayesian additive regression trees ( BART ). Statistics in Medicine, 35 0 (16): 0 2741--2753, 2016

2016
[13]

Bayesian A dditive R egression T rees with P arametric M odels of H eteroskedasticity

Justin Bleich and Adam Kapelner. Bayesian A dditive R egression T rees with P arametric M odels of H eteroskedasticity. arXiv preprint arXiv:1402.5397, 2014

2014 arXiv
[14]

Efficient M etropolis-- H astings proposal mechanisms for B ayesian regression tree models

Matthew T Pratola et al. Efficient M etropolis-- H astings proposal mechanisms for B ayesian regression tree models. B ayesian A nalysis , 11 0 (3): 0 885--911, 2016

2016
[15]

Log-linear B ayesian additive regression trees for multinomial logistic and count regression models

Jared S Murray. Log-linear B ayesian additive regression trees for multinomial logistic and count regression models. Journal of the American Statistical Association, 116 0 (534): 0 756--769, 2021

2021
[16]

Bart-based inference for poisson processes

Stamatina Lamprinakou, Mauricio Barahona, Seth Flaxman, Sarah Filippi, Axel Gandy, and Emma J McCoy. Bart-based inference for poisson processes. Computational Statistics & Data Analysis, 180: 0 107658, 2023

2023
[17]

Posterior concentration for B ayesian regression trees and forests

Veronika Ročková and Stéphanie van der Pas. Posterior concentration for B ayesian regression trees and forests . The Annals of Statistics, 48 0 (4): 0 2108 -- 2131, 2020

2020
[18]

On T heory for BART

Veronika Ro c kov\'a and Enakshi Saha. On T heory for BART . In Kamalika Chaudhuri and Masashi Sugiyama, editors, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pages 2...

2019
[19]

Bayesian regression tree ensembles that adapt to smoothness and sparsity

Antonio R Linero and Yun Yang. Bayesian regression tree ensembles that adapt to smoothness and sparsity. J ournal of the R oyal S tatistical S ociety: Series B ( S tatistical M ethodology) , 80 0 (5): 0 1087--1110, 2018

2018
[20]

Random tessellation forests

Shufei Ge, Shijia Wang, Yee Whye Teh, Liangliang Wang, and Lloyd Elliott. Random tessellation forests. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d Alch\' e -Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associ...

2019
[21]

The Ostomachion Process

Xuhui Fan, Bin Li, Yi Wang, Yang Wang, and Fang Chen. The Ostomachion Process . In Proceedings of the AAAI Conference on Artificial Intelligence, volume 30, 2016

2016
[22]

Sparse projection oblique randomer forests

Tyler M Tomita, James Browne, Cencheng Shen, Jaewon Chung, Jesse L Patsolic, Benjamin Falk, Carey E Priebe, Jason Yim, Randal Burns, Mauro Maggioni, et al. Sparse projection oblique randomer forests. Journal of Machine Learning Research, 21 0 (104), 2020

2020
[23]

Canonical correlation forests

Tom Rainforth and Frank Wood. Canonical correlation forests. arXiv preprint arXiv:1507.05444, 2015

2015 arXiv
[24]

Rotation forest: A new classifier ensemble method

Juan Jos \'e Rodriguez, Ludmila I Kuncheva, and Carlos J Alonso. Rotation forest: A new classifier ensemble method. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 0 (10): 0 1619--1630, 2006

2006
[25]

Regularizing axis-aligned ensembles via data rotations that favor simpler learners

Rico Blaser and Piotr Fryzlewicz. Regularizing axis-aligned ensembles via data rotations that favor simpler learners. Statistics and Computing, 2021

2021
[26]

Random rotation ensembles

Rico Blaser and Piotr Fryzlewicz. Random rotation ensembles. The Journal of Machine Learning Research, 2016

2016
[27]

Addivortes:(bayesian) additive voronoi tessellations

Adam J Stone and John Paul Gosling. Addivortes:(bayesian) additive voronoi tessellations. Journal of Computational and Graphical Statistics, 0 (just-accepted): 0 1--19, 2024

2024
[28]

Mateus Maia, Keefe Murphy, and Andrew C. Parnell. Gp-bart: A novel bayesian additive regression trees approach using gaussian processes. Computational Statistics & Data Analysis, 190: 0 107858, 2024. ISSN 0167-9473. doi:https://doi.org/10.1016/j.csda.2023.107858. URL https://w...

2024
[29]

Bamdt: Bayesian additive semi-multivariate decision trees for nonparametric regression

Zhao Tang Luo, Huiyan Sang, and Bani Mallick. Bamdt: Bayesian additive semi-multivariate decision trees for nonparametric regression. In International Conference on Machine Learning, pages 14509--14526. PMLR, 2022

2022
[30]

A survey of algorithms for geodesic paths and distances

Keenan Crane, Marco Livesu, Enrico Puppo, and Yipeng Qin. A survey of algorithms for geodesic paths and distances. arXiv preprint arXiv:2007.10430, 2020

2007 arXiv
[31]

Normalized cuts and image segmentation

Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Transactions on pattern analysis and machine intelligence, 22 0 (8): 0 888--905, 2000

2000
[32]

Diffusion maps

Ronald R Coifman and St \'e phane Lafon. Diffusion maps. Applied and computational harmonic analysis, 21 0 (1): 0 5--30, 2006

2006
[33]

Laplacian eigenmaps for dimensionality reduction and data representation

Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation, 15 0 (6): 0 1373--1396, 2003

2003
[34]

Random walks on graphs

F G \"o bel and AA Jagers. Random walks on graphs. Stochastic processes and their applications, 2 0 (4): 0 311--336, 1974

1974
[35]

Graph based gaussian processes on restricted domains

David B Dunson, Hau-Tieng Wu, and Nan Wu. Graph based gaussian processes on restricted domains. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84 0 (2): 0 414--439, 2022

2022
[36]

Understanding predictive information criteria for bayesian models

Andrew Gelman, Jessica Hwang, and Aki Vehtari. Understanding predictive information criteria for bayesian models. Statistics and computing, 24: 0 997--1016, 2014

2014
[37]

Stochastic tree ensembles for regularized nonlinear regression

Jingyu He and P Richard Hahn. Stochastic tree ensembles for regularized nonlinear regression. Journal of the American Statistical Association, 118 0 (541): 0 551--570, 2023

2023
[38]

The theory of branching processes, volume 6

Theodore Edward Harris et al. The theory of branching processes, volume 6. Springer Berlin, 1963

1963
[39]

Local gaussian process extrapolation for bart models with applications to causal inference

Meijia Wang, Jingyu He, and P Richard Hahn. Local gaussian process extrapolation for bart models with applications to causal inference. Journal of Computational and Graphical Statistics, 33 0 (2): 0 724--735, 2024

2024
[40]

B ayesian backfitting (with comments and a rejoinder by the authors)

Trevor Hastie and Robert Tibshirani. B ayesian backfitting (with comments and a rejoinder by the authors). Statistical Science, 15 0 (3): 0 196--223, 2000

2000
[41]

Strictly proper scoring rules, prediction, and estimation

Tilmann Gneiting and Adrian E Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American statistical Association, 102 0 (477): 0 359--378, 2007

2007
[42]

URL :https://geodacenter.github.io/data-and-lab/NYC_Tract_ACS2008_12/

G eo D a D ata and L ab. URL :https://geodacenter.github.io/data-and-lab/NYC_Tract_ACS2008_12/

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

[2] [2]

Bart: Bayesian additive regression trees

Hugh A Chipman, Edward I George, and Robert E McCulloch. Bart: Bayesian additive regression trees. 2010

2010

[3] [3]

A decision-theoretic generalization of on-line learning and an application to boosting

Yoav Freund and Robert E Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of computer and system sciences, 55 0 (1): 0 119--139, 1997

1997

[4] [4]

Bagging predictors

Leo Breiman. Bagging predictors. Machine learning, 24: 0 123--140, 1996

1996

[5] [5]

Random forests

Leo Breiman. Random forests. Machine learning, 45: 0 5--32, 2001

2001

[6] [6]

The B ayesian additive classification tree applied to credit risk modelling

Junni L Zhang and Wolfgang K H \"a rdle. The B ayesian additive classification tree applied to credit risk modelling. Computational Statistics & Data Analysis, 54 0 (5): 0 1197--1205, 2010

work page 2010

[7] [7]

Multinomial probit B ayesian additive regression trees

Bereket P Kindo, Hao Wang, and Edsel A Pe \ n a. Multinomial probit B ayesian additive regression trees. Stat, 5 0 (1): 0 119--131, 2016

work page 2016

[8] [8]

Variable selection for BART : an application to gene regulation

Justin Bleich, Adam Kapelner, Edward I George, Shane T Jensen, et al. Variable selection for BART : an application to gene regulation. T he A nnals of A pplied S tatistics , 8 0 (3): 0 1750--1781, 2014

work page 2014