Inference in popular nonparametric Bayesian models typically relies on sampling or other approximations. This paper presents a general methodology for constructing novel tractable nonparametric Bayesian methods by applying the kernel trick to inference in a parametric Bayesian model. For example, Gaussian process regre…
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48 results for “Bayesian nonparametric regression”
Bayesian methods estimate regression functions on submanifolds using graph Laplacian eigenbasis.
problem Estimating regression functions on unknown smooth submanifolds.
method Random geometric graph structure, Bayesian priors based on random basis expansion in graph Laplacian eigenbasis.
result Posterior contraction rates are minimax optimal for any positive smoothness index.
Causal Inference with the Instrumental Variable Approach and Bayesian Nonparametric Machine Learningstat.ML
Bayesian nonparametric machine learning improves instrumental variable inference.
problem Estimating causal effects with nonlinear relationships.
method Bayesian Additive Regression Trees (BART) for estimating functions and Dirichlet Process mixtures for error terms.
result Dramatic improvements in inference with nonlinear data, no manual tuning required.
This paper offers a simple method for Bayesian regression with unknown transformations.
problem Joint inference of unknown transformations and model parameters in Bayesian regression is computationally inefficient and cumbersome.
method The paper introduces a Bayesian nonparametric model via the Bayesian bootstrap to directly target the posterior distribution of the transformation.
result The approach delivers joint posterior consistency and efficient Monte Carlo inference for the transformation and all parameters.
The paper introduces a method for learning nonparametric Volterra kernels using Gaussian processes.
problem Learning nonparametric nonlinear operators from data.
method NVKM model using Volterra series and Gaussian processes for unobserved and observed input functions.
result The NVKM model can perform both single and multiple output regression and system identification.
Deep-HGP uses Bayesian nonparametric approach for complex data regression.
problem Complex data regression with compositional structures.
method Deep Gaussian processes with a squared-exponential kernel, data-driven lengthscale parameters.
result Posterior distribution optimally recovers unknown true regression curve in terms of quadratic loss.
Bayesian Additive Distribution Regression (DistBART) predicts distributions from grouped data.
problem Predicting distributions from grouped data with varying characteristics.
method Bayesian nonparametric approach using BART for modeling the regression function.
result Empirical and theoretical evidence supports DistBART's effectiveness in learning from low-dimensional marginals.
Bayesian nonparametric LABS model adapts to function smoothness in Besov spaces.
problem Estimating functions with unknown smoothness in Besov spaces.
method Lévy Adaptive B-spline (LABS) regression model with automatic smoothness adaptation.
result LABS posterior contracts around true function in Besov classes at nearly minimax-optimal rates.
Bayesian neural networks with nonparametric noise models for system identification.
problem Estimating parameters and noise processes in stochastic dynamic systems.
method Bayesian nonparametric approach using neural networks and Gibbs sampler.
result The method converges to full nonparametric Bayesian regression model.
Bayesian method with Gaussian process priors achieves optimal convergence rates for regression function and its derivatives.
problem Estimating the regression function and its derivatives in nonparametric regression.
method Bayesian approach with Gaussian process priors, focusing on convergence rates and plug-in property.
result Equivalence of convergence rates of posterior distributions and Bayes estimators for regression function and its derivatives.
Bayesian framework improves robustness in nonlinear regression models.
problem Measurement error, model misspecification, and distributional misspecification in regression analyses.
method Joint Dirichlet process prior on latent covariate-response distribution, updating with posterior pseudo-samples.
result Improved stability and consistency in estimators under increasing measurement error.
ASBART accelerates Soft BART for faster Bayesian regression.
problem Slow computation in Soft BART.
method Proposed ASBART, a variant of Soft BART.
result ASBART is about 10 times faster than Soft BART with similar accuracy.
SoftBart improves BART for high-noise modeling in science.
problem High noise in scientific data.
method Soft BART algorithm for Bayesian additive regression trees.
result Improves predictive performance and facilitates larger model integration.
This study converts BART to Gaussian process regression, revealing its limitations and potential improvements.
problem Understanding the Gaussian process limit of BART and its implications.
method Deriving and computing BART's prior covariance function, implementing the infinite trees limit as GP regression, and tuning hyperparameters.
result The Gaussian process limit of BART is inferior to standard BART but can be made competitive with proper hyperparameter tuning.
Bayesian Additive Regression Trees (BART) is a fully Bayesian approach to modeling with ensembles of trees. BART can uncover complex regression functions with high dimensional regressors in a fairly automatic way and provide Bayesian quantification of the uncertainty through the posterior. However, BART assumes IID nor…
Unlabeled Data Help in Graph-Based Semi-Supervised Learning: A Bayesian Nonparametrics Perspectivemath.ST
Bayesian analysis shows unlabeled data improve graph-based semi-supervised learning.
problem Improving semi-supervised learning with limited labeled data.
method Bayesian nonparametric approach using unlabeled data for graph-based learning.
result Posterior contracts optimally around the truth with sufficient unlabeled data.
Conjugate Projective Limitsmath.ST
We characterize conjugate nonparametric Bayesian models as projective limits of conjugate, finite-dimensional Bayesian models. In particular, we identify a large class of nonparametric models representable as infinite-dimensional analogues of exponential family distributions and their canonical conjugate priors. This c…
Nonparametric extension of tensor regression is proposed. Nonlinearity in a high-dimensional tensor space is broken into simple local functions by incorporating low-rank tensor decomposition. Compared to naive nonparametric approaches, our formulation considerably improves the convergence rate of estimation while maint…
GPU-accelerated BART speeds up Bayesian regression.
problem Long running time of BART makes it impractical for large datasets.
method GPU-enabled implementation of BART.
result BART is now 200x faster on GPUs.
Risk Margin Quantile Function Via Parametric and Non-Parametric Bayesian Quantile Regressionq-fin.RM
We develop quantile regression models in order to derive risk margin and to evaluate capital in non-life insurance applications. By utilizing the entire range of conditional quantile functions, especially higher quantile levels, we detail how quantile regression is capable of providing an accurate estimation of risk ma…
Additive nonparametric regression models provide an attractive tool for variable selection in high dimensions when the relationship between the response and predictors is complex. They offer greater flexibility compared to parametric non-linear regression models and better interpretability and scalability than the non-…
Bayesian posterior contraction rates improve with decreasing tails
problem Bayesian posterior contraction in nonparametric settings
method Using -exponential tails for contraction rates
result Improvement in contraction rates with decreasing tails
Quasi-experimental research designs, such as regression discontinuity and interrupted time series, allow for causal inference in the absence of a randomized controlled trial, at the cost of additional assumptions. In this paper, we provide a framework for discontinuity-based designs using Bayesian model comparison and …
We propose Bayesian extensions of two nonparametric regression methods which are kernel and mutual -nearest neighbor regression methods. Derived based on Gaussian process models for regression, the extensions provide distributions for target value estimates and the framework to select the hyperparameters. It is show…
To model categorical response variables given their covariates, we propose a permuted and augmented stick-breaking (paSB) construction that one-to-one maps the observed categories to randomly permuted latent sticks. This new construction transforms multinomial regression into regression analysis of stick-specific binar…
Increasingly complex datasets pose a number of challenges for Bayesian inference. Conventional posterior sampling based on Markov chain Monte Carlo can be too computationally intensive, is serial in nature and mixes poorly between posterior modes. Further, all models are misspecified, which brings into question the val…
Bayesian approach for estimating heterogeneous treatment effects in RDD designs.
problem Heterogeneity in treatment effects in RDD designs can lead to misleading conclusions.
method Direct Bayesian Additive Regression Trees (BART) for modeling heterogeneous treatment effects.
result Flexibly captures complicated structures of heterogeneous treatment effects as a function of covariates.
We propose a nonparametric Bayesian factor regression model that accounts for uncertainty in the number of factors, and the relationship between factors. To accomplish this, we propose a sparse variant of the Indian Buffet Process and couple this with a hierarchical model over factors, based on Kingman's coalescent. We…
Bayesian framework for sphere regression using Gaussian fields.
problem Nonparametric regression on the sphere with Gaussian priors.
method Isotropic Gaussian field priors, harmonic structure, exact posterior distributions, optimal spectral truncation, posterior contraction rates.
result Sharp posterior contraction rates for Gaussian priors with polynomially decaying angular power spectra.
Probit Monotone BARTstat.ML
Probit Monotone BART estimates binary outcomes using monotonic functions.
problem Estimating conditional mean functions for binary outcomes with monotonicity constraints.
method Proposes a new BART variant that incorporates monotonicity constraints for binary outcomes.
result Allows for more precise estimation of monotonic functions in binary outcome models.
In many applications, such as economics, operations research and reinforcement learning, one often needs to estimate a multivariate regression function f subject to a convexity constraint. For example, in sequential decision processes the value of a state under optimal subsequent decisions may be known to be convex or …
New method for high-dimensional linear regression using empirical Bayes.
problem Estimating prior in high-dimensional linear regression.
method Variational empirical Bayes approach with NPMLE and mean field approximation.
result Established asymptotic consistency and computational efficiency of the method.
Bayesian nonparametrics improves data-driven risk optimization under distributional uncertainty.
problem Improving out-of-sample performance in machine learning models due to distributional uncertainty.
method Combining Bayesian nonparametric theory and decision-theoretic preferences to propose a robust optimization criterion.
result The proposed robust optimization procedure provides favorable statistical guarantees and tractable approximations.
We propose Dirichlet Process mixtures of Generalized Linear Models (DP-GLM), a new method of nonparametric regression that accommodates continuous and categorical inputs, and responses that can be modeled by a generalized linear model. We prove conditions for the asymptotic unbiasedness of the DP-GLM regression mean fu…
Functional BART with Shape Priors: A Bayesian Tree Approach to Constrained Functional Regressionstat.ME
Functional BART adds shape priors to Bayesian tree regression for better curve fitting.
problem Regression with function-on-scalar data and shape constraints.
method Bayesian tree structure with spline representations, customized Bayesian backfitting algorithm, shape priors.
result Improved estimation and prediction accuracy with shape priors.
Bayesian deep learning with heavy-tailed weights achieves near-optimal performance.
problem Deep neural networks with heavy-tailed weights achieve near-optimal performance in various contexts.
method Introduced a Bayesian deep learning prior based on heavy-tailed weights and ReLU activation, showing near-optimal minimax contraction rates.
result Posterior distribution achieves near-optimal minimax contraction rates, adaptive to smoothness and intrinsic dimension.
Bayesian nonparametrics adapt model complexity to diverse datasets.
problem Complex challenges across statistics, computer science, and engineering.
method Flexible Bayesian nonparametric models that adapt model complexity.
result Bayesian nonparametrics offer innovative solutions to multi-object tracking.
The natural habitat of most Bayesian methods is data represented by exchangeable sequences of observations, for which de Finetti's theorem provides the theoretical foundation. Dirichlet process clustering, Gaussian process regression, and many other parametric and nonparametric Bayesian models fall within the remit of …
ABI bypasses likelihood intractability with nonparametric distribution matching.
problem Approximate Bayesian computation's inefficiency in high-dimensional settings and under diffuse priors.
method Adaptive Bayesian Inference (ABI) compares posterior distributions directly using nonparametric distribution matching and MSW distance.
result ABI significantly outperforms other methods in high-dimensional or dependent observation regimes.
A new model predicts multivariate regression using similarities to data points.
problem Complex, high-dimensional input-output relationships.
method Bayesian mixture-of-experts with conditional Gaussian mixtures and variational Bayes.
result Outperforms competitors in high-dimensional settings.
We develop a Bayesian "sum-of-trees" model where each tree is constrained by a regularization prior to be a weak learner, and fitting and inference are accomplished via an iterative Bayesian backfitting MCMC algorithm that generates samples from a posterior. Effectively, BART is a nonparametric Bayesian regression appr…
A Bayesian Nonparametric Perspective on Mahalanobis Distance for Out of Distribution Detectionstat.ML
Bayesian nonparametric models improve OOD detection, especially with complex covariance structures.
problem Improving out-of-distribution detection methods, especially in complex scenarios.
method Proposes Bayesian nonparametric mixture models with hierarchical priors that generalize the Mahalanobis distance score.
result Bayesian nonparametric methods outperform existing OOD methods, especially in complex scenarios.
It is now known that an extended Gaussian process model equipped with rescaling can adapt to different smoothness levels of a function valued parameter in many nonparametric Bayesian analyses, offering a posterior convergence rate that is optimal (up to logarithmic factors) for the smoothness class the true function be…
Tree-based synthesis improves forecast accuracy in GDP and inflation.
problem Improving forecast accuracy in GDP and inflation.
method Developed a nonparametric synthesis function using regression trees.
result Tree-based synthesis leads to improved forecast accuracy.
From Partial Exchangeability to Predictive Probability: A Bayesian Perspective on Classificationstat.ME
Bayesian model improves classification performance with flexible uncertainty modeling.
problem Improving classification performance with flexible uncertainty modeling.
method Combines Gaussian process and Dirichlet process priors for latent function and link function, respectively.
result Outperforms standard logistic regression on simulated data.
To restore the historical sea surface temperatures (SSTs) better, it is important to construct a good calibration model for the associated proxies. In this paper, we introduce a new model for alkenone () based on the heteroscedastic Gaussian process (GP) regression method. Our nonparametric app…
New rigorous uncertainty bounds for Gaussian Process regression.
problem Need for frequentist uncertainty bounds in applications like learning-based control.
method Introduce new uncertainty bounds that are rigorous and practically useful.
result New bounds are less conservative and more useful for practical applications.
We introduce the nonparametric metadata dependent relational (NMDR) model, a Bayesian nonparametric stochastic block model for network data. The NMDR allows the entities associated with each node to have mixed membership in an unbounded collection of latent communities. Learned regression models allow these memberships…