Pith. sign in

REVIEW 3 major objections 5 minor 300 references

Sparse Bayesian networks let high-dimensional distribution estimation beat the curse of dimensionality.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-30 15:54 UTC pith:LLHAK7VZ

load-bearing objection Clean sparse-BN + regression estimator with honest TV rates under known parents; the theory–practice gap on structure recovery is real but clearly flagged, not hidden. the 3 major comments →

arxiv 2607.26955 v1 pith:LLHAK7VZ submitted 2026-07-29 stat.ML cs.LGstat.ME

Breaking the Curse with BAND: Nonparametric Distribution Estimation in High Dimensions

Shuo-Chieh Huang , Chien-Ming Chi , Jau-er Chen This is my paper
classification stat.ML cs.LGstat.ME MSC 62G0762H1262M10
keywords distribution estimationcurse of dimensionalityBayesian networkssparse regressiontotal variationmixed-type datatime serieshigh-dimensional statistics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Classical nonparametric density estimation slows exponentially as dimension grows, so total-variation rates become useless once the number of variables exceeds a few logs of sample size. This paper shows that if the joint law factors as a sparse Bayesian network—each coordinate depending on only a few predecessors—then estimating those conditionals with ordinary sparse regression (trees, Lasso, boosting) yields a total-variation error that shrinks like a polynomial in sample size even while dimension itself grows polynomially. The resulting estimator, BAND, works for continuous, discrete and mixed data and for stationary time series under mild mixing. Empirically it matches or beats specialized generative models on sampling and multivariate forecast regions. The practical payoff is a usable high-dimensional distribution model whose rates no longer require the ambient dimension to stay tiny.

Core claim

Under a sparse Bayesian-network factorization (each variable depends on at most s0 earlier variables) and strong mixing, the BAND estimator attains a total-variation rate of order p n^{-1/(3+s0)} (up to mild logarithmic and binning factors) while the dimension p may grow polynomially with n—orders of magnitude faster than the classical minimax histogram rate that forces p = o(log n).

What carries the argument

BAND: discretize each coordinate into bins, encode the bins as one-hot vectors, then estimate every conditional probability by a sparsity-aware mean regression (tree, Lasso or CatBoost); the product of these estimated conditionals reconstitutes the joint law on the resulting hypercubes.

Load-bearing premise

The proved rates treat the true parent sets of each variable as known; learning those sets from data is left open, so the theoretical guarantee does not automatically cover the black-box regression used in practice.

What would settle it

Generate data from a dense Bayesian network (every variable depends on all predecessors) with p growing like n^α for α>0 and check whether BAND’s total-variation error still decays polynomially; if the error plateaus or worsens, the sparsity claim is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • High-dimensional mixed-type time series can be sampled and density-estimated at rates independent of ambient dimension once sparsity is present.
  • Multivariate forecast confidence regions become computationally and statistically feasible for dozens of macroeconomic series.
  • Any sparse conditional-mean method (Lasso, trees, bounded-fan-in nets) can be plugged in to obtain a joint distribution estimator with matching rates.
  • Discrete high-dimensional distributions with sparse dependence escape the exponential support-size barrier that otherwise dominates minimax rates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If parent-set recovery can be shown at the same rate, the entire pipeline becomes fully data-driven and the theory would cover the CatBoost implementation used in the experiments.
  • The same sparse-network idea could be ported to score-based diffusion models to obtain high-dimensional rates without Barron-space or low-dimensional-support assumptions.
  • BAND’s hypercube representation offers a natural route to conditional generative modeling and missing-value imputation under sparsity.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes BAND, a distribution estimator that factors a joint law through a Bayesian network and estimates each conditional probability by sparsity-aware regression on one-hot bin indicators. Under a sparse parent-set assumption (max |Sj| ≤ s0), strong mixing, and a score bound, Theorem 3.3 and Corollary 3.7 give a total-variation rate of order p n^{-1/(3+s0)} (up to logs and bin geometry) while allowing p to grow polynomially in n—faster than classical histogram rates that require p = o(log n). Discrete data yield a linear-in-p rate (Corollary 3.9). The method handles mixed types and stationary time series. Simulations (Gaussian/uniform mixtures, block-Gaussian) and a macro forecasting application show competitive Cramér–Wold distances and coverage gaps versus mclust, vine copulas, normalizing flows, QR, and QRF. Code is released.

Significance. If the rates transfer to a fully data-driven procedure, the contribution is substantial: a practical, mixed-type, time-series-compatible estimator with polynomial-in-p TV rates under a transparent sparsity condition, together with detailed bias–variance and mixing arguments (Theorem B.1, Lemma B.2, Corollaries 3.7–3.9) and reproducible experiments. The oracle analysis already clarifies how sparse BN structure replaces ambient dimension in the exponent, aligning with and extending recent sample-complexity results (e.g., Vandermeulen et al.). The gap between oracle theory and the CatBoost implementation is the main caveat; closing or clearly delimiting it would make the paper a solid reference for high-dimensional distribution modeling.

major comments (3)
  1. [§3.2, Condition 3.5, Eq. (4), Corollary 3.7, Example 3.8] Corollary 3.7 / Example 3.8 and the tree estimator (4) treat the parent sets Sj of Condition 3.5 as known. The manuscript explicitly leaves recovery of Sj to future work (p. 4). All experiments and the released implementation fit conditionals with CatBoost, for which there is no high-probability guarantee that the learned parents coincide with the true skeleton or that the plug-in product still satisfies the ep√T/dmin term of Theorem 3.3. The abstract and introduction attribute the polynomial-in-p rates to BAND itself. Either supply a structure-recovery analysis (e.g., Lasso/CART consistency under the same mixing and sparsity conditions) that closes the gap, or rewrite the claims so that the proved rates are clearly labeled as oracle rates and the empirical method is evaluated separately.
  2. [Example 3.8, Corollary 3.7, Corollary 3.9] The continuous rate in Example 3.8 is p n^{-1/(3+s0+2b0)} (up to logs). The linear factor of p remains; when s0 is moderate and p is large this can still be slower than methods that achieve rates depending only on intrinsic dimension or graph resilience without an explicit p multiplier. The comparison to the classical histogram rate n^{-β/(2β+p)} is correct for consistency when p ≫ log n, but the paper should state more carefully for which (p, s0, n) regimes the BAND bound is numerically superior, and whether the p factor is an artifact of the union bound over coordinates (cf. the discrete case in Corollary 3.9, which is also linear in p).
  3. [§4.2, Tables 1–3; §5, Table 5; Appendix A.1–A.2] Empirical evaluation uses the Cramér–Wold distance (Appendix A.1), a computable lower-bound surrogate for TV, not TV itself. Tables 1–3 therefore do not directly corroborate the TV rates. In addition, SPTF and the binning factor are tuned on the same tasks used for reporting (CW minimization; coverage calibration on the 6-month horizon). A clearer protocol—held-out tuning, or reporting sensitivity to factor/SPTF—would strengthen the claim that BAND “performs competitively” in a way that supports the theoretical narrative.
minor comments (5)
  1. [§3.2, Condition 3.5(a); §D.2] Condition 3.5(a) is imposed on the discretized one-hot vectors, not on the original continuous variables. The text notes this and gives Example 4.2 / §D.2 as justification, but a short remark in §3.2 on when fine binning preserves the continuous sparse BN would help readers.
  2. [§2.2–§3] Notation switches between bejl, bμ, eμ, and various boldface conventions; a small notation table or consistent use of hats would improve readability in §2–3 and the supplement.
  3. [Figure 2, Tables 1–3] Figure 2 is only for p=2; a higher-dimensional visual or marginal diagnostic for the p=30 cases in Tables 1–3 would make the sampling quality easier to assess.
  4. [§1] The GitHub link uses a placeholder name (no-name213); ensure the camera-ready version points to a stable repository with seeds and scripts that regenerate Tables 1–5.
  5. [Abstract, §4.2] Typos / style: “BAyesian” capitalization in the acronym expansion; “nflowsin” missing space (p. 6); occasional double spaces and “the the”.

Circularity Check

0 steps flagged

No significant circularity: TV rates are derived from external sparsity, mixing, and score assumptions via an oracle tree estimator, not from quantities defined in terms of the estimator itself.

full rationale

The load-bearing chain is Theorem 3.3 (bias–variance bound under Conditions 3.1–3.2) specialized in Corollaries 3.7/3.9 under sparse Bayesian networks (Condition 3.5), strong mixing (Condition 3.6), and bin geometry. The deviation √T and lower bound dmin are obtained from concentration of the oracle sparse-tree estimator (4) that conditions on the true parent sets Sj; those sets are assumed known for the rate proofs and recovery is explicitly deferred. Empirical knobs (factor, SPTF) affect only finite-sample experiments and do not enter the asymptotic statements. Citations (Vandermeulen et al., standard mixing/inequality references, CatBoost) supply external tools or comparable rates; none is a self-citation that forces the claimed polynomial-in-p rate by construction. The theory–practice gap (oracle Sj vs. CatBoost) is a correctness/transfer issue, not circularity. No equation equates the claimed rate to a fitted constant or renames an input as a prediction.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central rate claim rests on a short list of structural and regularity assumptions plus a few practical tuning knobs. The load-bearing modeling axiom is the sparse Bayesian-network factorization with known parent sets in the theorems; regularity axioms are a uniform score bound, geometric strong mixing of the relevant subprocesses, and density bounds on small coordinate subsets. Free parameters appear only in the finite-sample implementation (bin factor, SPTF, CatBoost learning rate). No new physical entities are postulated.

free parameters (3)
  • binning factor (number of bins ~ factor × n^{1/3}) = 2 (sims); 0.9 / 1.0 (two forecast periods)
    Chosen as 2 in simulations; selected on a grid {0.4,…,2.0} by coverage calibration in the macro application (Table 4).
  • SPTF (sampling probability threshold factor) = tuned per experiment in [0,0.4]
    Threshold in [0,0.4] that zeros low conditional bin probabilities at sampling time; tuned by minimizing empirical Cramér–Wold distance to training data.
  • CatBoost learning rate = 0.05
    Fixed at 0.05 after informal stability checks; other CatBoost hyperparameters left at library defaults.
axioms (6)
  • domain assumption Condition 3.5: each discretized (or original) coordinate is conditionally independent of non-parents given at most s0 parents in a directed order.
    Load-bearing sparsity model that replaces ambient dimension p by s0 in the rate; invoked for Corollaries 3.7 and 3.9.
  • ad hoc to paper Parent sets Sj are treated as known when stating the tree estimator (4) and the rates.
    Explicitly deferred to future work; creates a theory–practice gap because the implemented method estimates conditionals with CatBoost without a proved parent-recovery guarantee.
  • domain assumption Condition 3.2: log-density (or conditional density) has gradient bounded by K1√p R on the compact [−δn,δn]^p.
    Standard score regularity used to control histogram bias; verified for Gaussian, Student-t, and boundary-adjusted uniforms and their mixtures in §D.1.
  • domain assumption Condition 3.6: each relevant subprocess is geometrically strongly mixing.
    Needed for the Bernstein-type concentration of the sparse tree estimator under dependence.
  • domain assumption Joint densities of every subset of size ≤ s0+1 are bounded above and below by positive constants (continuous case), or cell probabilities satisfy geometric bounds (discrete case).
    Produces the dmin lower bound in Condition 3.1 that keeps the product of estimated conditionals stable.
  • standard math Standard facts on α-mixing covariance inequalities and histogram bias under Lipschitz scores.
    Rio (1993), Merlevède et al. (2009), classical MVT-for-integrals arguments used in §§B.2–B.4.

pith-pipeline@v1.2.0-daily-grok45 · 37125 in / 3715 out tokens · 88496 ms · 2026-07-30T15:54:20.554744+00:00 · methodology

0 comments
read the original abstract

Minimax-optimal rates for multivariate distribution estimation are known to suffer from the curse of dimensionality. We propose a sparse Bayesian network approach in which each conditional probability is estimated using sparsity-aware conditional mean methods. The resulting estimator, \textit{BAyesian Network Distribution regression} (BAND), handles mixed data types in high-dimensional time series and achieves polynomial total variation convergence rates while allowing the feature dimension to grow polynomially with the sample size. These rates are substantially faster than the classical optimal rates for multivariate histogram density estimators that lack sparsity. Empirical evaluations show that BAND performs competitively for data sampling and confidence region forecasting against a range of state-of-the-art benchmarks.

Figures

Figures reproduced from arXiv: 2607.26955 by Chien-Ming Chi, Jau-er Chen, Shuo-Chieh Huang.

Figure 1
Figure 1. Figure 1: (Left) The two-dimensional feature space is par￾titioned into four regions due to discrete points A and B, including three “discrete” regions indexed by Θ = {({1}, A),({2}, B),({1, 2},(A, B))} and the rest correspond￾ing to the continuous part. (Right) Hypercube partition of the feature support and bins along each coordinate. The discrete points A and B are themselves bins. There are three bins along the f… view at source ↗
Figure 2
Figure 2. Figure 2: Scatterplots of 10,000 empirical data points (blue) and 10,000 points sampled from the four fitted models (red). Data are drawn from Example 4.2 with r = 4. Each method is trained on n = 1000 samples with p = 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

300 extracted references · 26 linked inside Pith

  1. [1]

    Biometrika , volume=

    An alternative method of cross-validation for the smoothing of density estimates , author=. Biometrika , volume=. 1984 , publisher=

    1984

  2. [2]

    arXiv preprint arXiv:2412.06220 , year=

    Sliced Distribution Matching based on Cumulative Distribution Functions with Applications to Control , author=. arXiv preprint arXiv:2412.06220 , year=

    arXiv

  3. [3]

    The Journal of Machine Learning Research , volume=

    Analysis of a random forests model , author=. The Journal of Machine Learning Research , volume=. 2012 , publisher=

    2012

  4. [4]

    Journal of Machine Learning Research , volume=

    Cramer-wold auto-encoder , author=. Journal of Machine Learning Research , volume=

  5. [5]

    Journal of Nonparametric Statistics , volume=

    On the lrerror in histogram density estimation: The multidimensional case , author=. Journal of Nonparametric Statistics , volume=. 1998 , publisher=

    1998

  6. [6]

    Advances in Neural Information Processing Systems , volume=

    Breaking the curse of dimensionality in structured density estimation , author=. Advances in Neural Information Processing Systems , volume=

  7. [7]

    arXiv preprint arXiv:2501.02298 , year=

    Beyond log-concavity and score regularity: Improved convergence bounds for score-based generative models in w2-distance , author=. arXiv preprint arXiv:2501.02298 , year=

    arXiv

  8. [8]

    arXiv preprint arXiv:2503.12966 , year=

    Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity , author=. arXiv preprint arXiv:2503.12966 , year=

    arXiv

  9. [9]

    Statistics & Probability Letters , volume=

    A sharp estimate of the binomial mean absolute deviation with applications , author=. Statistics & Probability Letters , volume=. 2013 , publisher=

    2013

  10. [10]

    2017 , publisher=

    Classification and regression trees , author=. 2017 , publisher=

    2017

  11. [11]

    Journal of the American Statistical Association , volume=

    Problems in the analysis of survey data, and a proposal , author=. Journal of the American Statistical Association , volume=

  12. [12]

    International Conference on Machine Learning (ICML) , year =

    Conformal prediction for multi-dimensional time series by ellipsoidal sets , author=. International Conference on Machine Learning (ICML) , year =

  13. [13]

    Advances in Neural Information Processing Systems (NeurIPS) , volume=

    Conformal time-series forecasting , author=. Advances in Neural Information Processing Systems (NeurIPS) , volume=

  14. [14]

    International Conference on Learning Representations (ICLR) , year=

    Copula conformal prediction for multi-step time series prediction , author=. International Conference on Learning Representations (ICLR) , year=

  15. [15]

    The American Statistician , volume=

    Computing and graphing highest density regions , author=. The American Statistician , volume=. 1996 , publisher=

    1996

  16. [16]

    Communications in Statistics-Simulation and Computation , volume=

    Confidence intervals for quantiles from histograms and other grouped data , author=. Communications in Statistics-Simulation and Computation , volume=. 2020 , publisher=

    2020

  17. [17]

    , author=

    Quantile regression forests. , author=. Journal of Machine Learning Research , volume=

  18. [18]

    International Conference on Machine Learning (ICML) , year=

    Training diffusion-based generative models with limited data , author=. International Conference on Machine Learning (ICML) , year=

  19. [19]

    Biometrika , volume=

    On optimal and data-based histograms , author=. Biometrika , volume=. 1979 , publisher=

    1979

  20. [20]

    doi:10.5281/zenodo.4296287 , url =

    Conor Durkan and Artur Bekasov and Iain Murray and George Papamakarios , title =. doi:10.5281/zenodo.4296287 , url =

    doi:10.5281/zenodo.4296287

  21. [21]

    Advances in Neural Information Processing Systems (NeurIPS) , year=

    Neural Spline Flows , author=. Advances in Neural Information Processing Systems (NeurIPS) , year=

  22. [22]

    International Conference on Learning Representations (ICLR) , year=

    Score-Based Generative Modeling through Stochastic Differential Equations , author=. International Conference on Learning Representations (ICLR) , year=

  23. [23]

    On the use of

    Juan, Alfons and Vidal, Enrique , journal=. On the use of. 2002 , publisher=

    2002

  24. [24]

    Statistics Surveys , pages =

    Volodymyr Melnykov and Ranjan Maitra , title =. Statistics Surveys , pages =. 2010 , doi =

    2010

  25. [25]

    Annual review of statistics and its application , volume=

    Finite mixture models , author=. Annual review of statistics and its application , volume=. 2019 , publisher=

    2019

  26. [26]

    Journal of the London Mathematical Society , volume=

    Some theorems on distribution functions , author=. Journal of the London Mathematical Society , volume=. 1936 , publisher=

    1936

  27. [27]

    Machine Intelligence Research , pages=

    Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models , author=. Machine Intelligence Research , pages=. 2025 , publisher=

    2025

  28. [28]

    Journal of the American Statistical Association , volume=

    Prediction of spatial cumulative distribution functions using subsampling , author=. Journal of the American Statistical Association , volume=. 1999 , publisher=

    1999

  29. [29]

    Brendan and Raftery, Adrian E

    Scrucca, Luca and Fop, Michael and Murphy, T. Brendan and Raftery, Adrian E. , journal =. mclust 5: clustering, classification and density estimation using

  30. [30]

    Journal of Multivariate Analysis , volume=

    Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas , author=. Journal of Multivariate Analysis , volume=. 2016 , publisher=

    2016

  31. [31]

    Zeitschrift f

    On the histogram as a density estimator: L 2 theory , author=. Zeitschrift f. 1981 , publisher=

    1981

  32. [32]

    International Conference on Artificial Intelligence and Statistics , pages=

    Langevin monte carlo without smoothness , author=. International Conference on Artificial Intelligence and Statistics , pages=. 2020 , organization=

    2020

  33. [33]

    Journal of the American statistical Association , volume=

    Strictly proper scoring rules, prediction, and estimation , author=. Journal of the American statistical Association , volume=. 2007 , publisher=

    2007

  34. [34]

    International Conference on Learning Representations (ICLR) , year=

    PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications , author=. International Conference on Learning Representations (ICLR) , year=

  35. [35]

    Finite state

    Tauchen, George , journal=. Finite state

  36. [36]

    Friedman, Nir and Linial, Michal and Nachman, Iftach and Pe'er, Dana , booktitle=. Using

  37. [37]

    Journal of Machine Learning Research , volume=

    A neural probabilistic language model , author=. Journal of Machine Learning Research , volume=

  38. [38]

    International Conference on Machine Learning (ICML) , year=

    Diffusion models are minimax optimal distribution estimators , author=. International Conference on Machine Learning (ICML) , year=

  39. [39]

    arXiv preprint arXiv:2408.12863 , year=

    Machine learning and the yield curve: Tree-based macroeconomic regime switching , author=. arXiv preprint arXiv:2408.12863 , year=

    Pith/arXiv arXiv

  40. [40]

    BMC genomics , volume=

    Genome-wide association data classification and SNPs selection using two-stage quality-based Random Forests , author=. BMC genomics , volume=. 2015 , publisher=

    2015

  41. [41]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Multi-scale domain-adversarial multiple-instance CNN for cancer subtype classification with unannotated histopathological images , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

  42. [42]

    International Conference on Learning Representations (ICLR) , year=

    Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian distributions , author=. International Conference on Learning Representations (ICLR) , year=

  43. [43]

    IEEE Transactions on Information Theory , volume=

    Efficient agnostic learning of neural networks with bounded fan-in , author=. IEEE Transactions on Information Theory , volume=. 1996 , publisher=

    1996

  44. [44]

    Journal of the American Statistical Association , volume=

    Forecasting using principal components from a large number of predictors , author=. Journal of the American Statistical Association , volume=

  45. [45]

    2009 , series =

    Introduction to Nonparametric Estimation , author =. 2009 , series =

    2009

  46. [46]

    arXiv preprint arXiv:2408.05807 , year=

    Kernel density estimators in large dimensions , author=. arXiv preprint arXiv:2408.05807 , year=

    Pith/arXiv arXiv

  47. [47]

    arXiv preprint arXiv:2402.15602 , year=

    Minimax optimality of score-based diffusion models: Beyond the density lower bound assumptions , author=. arXiv preprint arXiv:2402.15602 , year=

    Pith/arXiv arXiv

  48. [48]

    Proceedings of the National Academy of Sciences , volume=

    Sampling with flows, diffusion, and autoregressive neural networks from a spin-glass perspective , author=. Proceedings of the National Academy of Sciences , volume=. 2024 , publisher=

    2024

  49. [49]

    Zhang , title =

    Sitan Chen and Sinho Chewi and Jerry Li and Yuanzhi Li and Adil Salim and Anru R. Zhang , title =. International Conference on Learning Representations (ICLR) , year =

  50. [50]

    Wiley Interdisciplinary Reviews: Computational Statistics , volume=

    Copulae: An overview and recent developments , author=. Wiley Interdisciplinary Reviews: Computational Statistics , volume=. 2022 , publisher=

    2022

  51. [51]

    Journal of Econometrics , volume=

    Stationary vine copula models for multivariate time series , author=. Journal of Econometrics , volume=. 2022 , publisher=

    2022

  52. [52]

    International Conference on Artificial Intelligence and Statistics , pages=

    Adversarial random forests for density estimation and generative modeling , author=. International Conference on Artificial Intelligence and Statistics , pages=. 2023 , organization=

    2023

  53. [53]

    Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining , pages=

    Density estimation trees , author=. Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining , pages=

  54. [54]

    Advances in Neural Information Processing Systems (NeurIPS) , volume=

    Density estimation via discrepancy based adaptive sequential partition , author=. Advances in Neural Information Processing Systems (NeurIPS) , volume=

  55. [55]

    Journal of Business & Economic Statistics , volume=

    Modeling dependence in high dimensions with factor copulas , author=. Journal of Business & Economic Statistics , volume=. 2017 , publisher=

    2017

  56. [56]

    Journal of the American Statistical Association , volume=

    Multivariate density estimation by bayesian sequential partitioning , author=. Journal of the American Statistical Association , volume=. 2013 , publisher=

    2013

  57. [57]

    Journal of Machine Learning Research , volume=

    Convergence rates of a class of multivariate density estimation methods based on adaptive partitioning , author=. Journal of Machine Learning Research , volume=

  58. [58]

    Handbook of statistics , volume=

    Multidimensional density estimation , author=. Handbook of statistics , volume=. 2005 , publisher=

    2005

  59. [59]

    Journal of statistical planning and inference , volume=

    Density estimation for time series by histograms , author=. Journal of statistical planning and inference , volume=. 1994 , publisher=

    1994

  60. [60]

    The Annals of Statistics , volume=

    On bandwidth choice for density estimation with dependent data , author=. The Annals of Statistics , volume=. 1995 , publisher=

    1995

  61. [61]

    Journal of the American Statistical Association , volume=

    Rectangular confidence regions for the means of multivariate normal distributions , author=. Journal of the American Statistical Association , volume=

  62. [62]

    Statistica Sinica , pages=

    The geometrical ergodicity of nonlinear autoregressive models , author=. Statistica Sinica , pages=. 1996 , publisher=

    1996

  63. [63]

    2005 , publisher=

    Analysis of financial time series , author=. 2005 , publisher=

    2005

  64. [64]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Regression shrinkage and selection via the lasso , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 1996 , publisher=

    1996

  65. [65]

    Annales de l'IHP Probabilit

    Covariance inequalities for strongly mixing processes , author=. Annales de l'IHP Probabilit

  66. [66]

    High dimensional probability V: the Luminy volume , volume=

    Bernstein inequality and moderate deviations under strong mixing conditions , author=. High dimensional probability V: the Luminy volume , volume=. 2009 , publisher=

    2009

  67. [67]

    Journal of Official Statistics , volume=

    A note on choice of bivariate histogram bin shape , author=. Journal of Official Statistics , volume=. 1988 , publisher=

    1988

  68. [68]

    Journal of Financial economics , volume=

    A general approach to integrated risk management with skewed, fat-tailed risks , author=. Journal of Financial economics , volume=. 2006 , publisher=

    2006

  69. [69]

    Nonparametric Density Estimation: The L_1 View , series =

    Luc Devroye and L. Nonparametric Density Estimation: The L_1 View , series =. 1985 , address =

    1985

  70. [70]

    Wiley Interdisciplinary Reviews: Computational Statistics , volume=

    Nonparametric density estimation for high-dimensional data—Algorithms and applications , author=. Wiley Interdisciplinary Reviews: Computational Statistics , volume=. 2019 , publisher=

    2019

  71. [71]

    2015 , publisher=

    Multivariate density estimation: theory, practice, and visualization , author=. 2015 , publisher=

    2015

  72. [72]

    The Annals of Statistics , volume=

    Consistency of data-driven histogram methods for density estimation and classification , author=. The Annals of Statistics , volume=. 1996 , publisher=

    1996

  73. [73]

    Statistica sinica , pages=

    Multivariate histograms with data-dependent partitions , author=. Statistica sinica , pages=. 2009 , publisher=

    2009

  74. [74]

    Journal of Machine Learning Research , volume=

    Distributional random forests: Heterogeneity adjustment and multivariate distributional regression , author=. Journal of Machine Learning Research , volume=

  75. [75]

    2012 , publisher=

    Machine learning: a probabilistic perspective , author=. 2012 , publisher=

    2012

  76. [76]

    Journal of Multivariate Analysis , volume=

    A review of copula models for economic time series , author=. Journal of Multivariate Analysis , volume=. 2012 , publisher=

    2012

  77. [77]

    Computational Statistics & Data Analysis , volume=

    Multivariate locally adaptive density estimation , author=. Computational Statistics & Data Analysis , volume=. 2002 , publisher=

    2002

  78. [78]

    DISTRIBUTIONAL REGRESSION FORESTS FOR PROBABILISTIC PRECIPITATION FORECASTING IN COMPLEX TERRAIN , urldate =

    Lisa Schlosser and Torsten Hothorn and Reto Stauffer and Achim Zeileis , journal =. DISTRIBUTIONAL REGRESSION FORESTS FOR PROBABILISTIC PRECIPITATION FORECASTING IN COMPLEX TERRAIN , urldate =

  79. [79]

    Journal of Econometrics , volume=

    Cross-validated SNP density estimates , author=. Journal of Econometrics , volume=. 2002 , publisher=

    2002

  80. [80]

    Journal of the American Statistical Association , volume=

    Methods for estimating a conditional distribution function , author=. Journal of the American Statistical Association , volume=

Showing first 80 references.