Regularization in fitting regression models has been a highly active topic of research in the past few decades, but most of the existing methods are designed for particular situations, e.g. for the case of a sparse coefficient vector. We consider the problem of designing $\textit{universally}$ optimal regularized estimators in a given generalized linear model with fixed effects. First, we propose as a contender the Bayes estimator against an $\textit{ideal}$ prior that assigns equal mass to every permutation of the fixed coefficient vector, thus depending on the true coefficients only through their empirical CDF. We prove some optimality properties of this oracle estimator in both the frequentist and Bayesian frameworks. To compete with the oracle estimator, we posit a hierarchical Bayes model where the individual coefficients are modeled as i.i.d. draws from a common distribution $π$, which is in turn assigned a Polya tree prior to reflect indefiniteness. We demonstrate in examples that the posterior mean of $π$ under the postulated model adapts nonparametrically to the empirical CDF of the true coefficients. Correspondingly, the posterior means of the coefficients themselves are used to mimic the ideal estimator. Numerical experiments show that our method has better estimation and prediction accuracy compared to various parametric and nonparametric alternatives, from relatively standard $L_p$-regularized estimators to modern penalized-likelihood and Bayesian estimators for high dimensional regression.
Wavelet shrinkage estimators are widely applied in several fields of science for denoising data in wavelet domain by reducing the magnitudes of empirical coefficients. In nonparametric regression problem, most of the shrinkage rules are derived from models composed by an unknown function with additive gaussian noise. Although gaussian noise assumption is reasonable in several real data analysis, mainly for large sample sizes, it is not general. Contaminated data with positive noise can occur in practice and nonparametric regression models with positive noise bring challenges in wavelet shrinka…
Bayesian nonparametric regression with dependent wavelets has dual shrinkage properties: there is shrinkage through a dependent prior put on functional differences, and shrinkage through the setting of most of the wavelet coefficients to zero through Bayesian variable selection methods. The methodology can deal with unequally spaced data and is efficient because of the existence of fast moves in model space for the MCMC computation. The methodology is illustrated on the problem of modeling the oscillations of Cepheid variable stars; these are a class of pulsating variable stars with the useful…
We present a locally adaptive nonparametric curve fitting method that operates within a fully Bayesian framework. This method uses shrinkage priors to induce sparsity in order-k differences in the latent trend function, providing a combination of local adaptation and global control. Using a scale mixture of normals representation of shrinkage priors, we make explicit connections between our method and kth order Gaussian Markov random field smoothing. We call the resulting processes shrinkage prior Markov random fields (SPMRFs). We use Hamiltonian Monte Carlo to approximate the posterior distri…
This work studies the variation in Kullback-Leibler divergence between random draws from some popular nonparametric processes and their baseline measure. In particular we focus on the Dirichlet process, the Pólya tree and the frequentist and Bayesian bootstrap. The results shed light on the support of these nonparametric processes. Of particular note are results for finite Pólya trees that are used to model continuous random probability measures. Our results provide guidance for specifying the parameterisation of the Pólya tree process that allows for greater understanding while highlighting l…
In a remarkable series of papers beginning in 1956, Charles Stein set the stage for the future development of minimax shrinkage estimators of a multivariate normal mean under quadratic loss. More recently, parallel developments have seen the emergence of minimax shrinkage estimators of multivariate normal predictive densities under Kullback--Leibler risk. We here describe these parallels emphasizing the focus on Bayes procedures and the derivation of the superharmonic conditions for minimaxity as well as further developments of new minimax shrinkage predictive density estimators including mult…
We consider nonparametric Bayesian estimation and prediction for nonhomogeneous Poisson process models with unknown intensity functions. We propose a class of improper priors for intensity functions. Nonparametric Bayesian inference with kernel mixture based on the class improper priors is shown to be useful, although improper priors have not been widely used for nonparametric Bayes problems. Several theorems corresponding to those for finite-dimensional independent Poisson models hold for nonhomogeneous Poisson process models with infinite-dimensional parameter spaces. Bayesian estimation and…
Nonparametric and nonlinear measures of statistical dependence between pairs of random variables are important tools in modern data analysis. In particular the emergence of large data sets can now support the relaxation of linearity assumptions implicit in traditional association scores such as correlation. Here we describe a Bayesian nonparametric procedure that leads to a tractable, explicit and analytic quantification of the relative evidence for dependence vs independence. Our approach uses Polya tree priors on the space of probability measures which can then be embedded within a decision …
We propose a novel class of dynamic shrinkage processes for Bayesian time series and regression analysis. Building upon a global-local framework of prior construction, in which continuous scale mixtures of Gaussian distributions are employed for both desirable shrinkage properties and computational tractability, we model dependence among the local scale parameters. The resulting processes inherit the desirable shrinkage behavior of popular global-local priors, such as the horseshoe prior, but provide additional localized adaptivity, which is important for modeling time series data or regressio…
Bayesian hierarchical models are used to share information between related samples and obtain more accurate estimates of sample-level parameters, common structure, and variation between samples. When the parameter of interest is the distribution or density of a continuous variable, a hierarchical model for continuous distributions is required. A number of such models have been described in the literature using extensions of the Dirichlet process and related processes, typically as a distribution on the parameters of a mixing kernel. We propose a new hierarchical model based on the Pólya tree, …
A new class of dependent random measures which we call {\it compound random measures} are proposed and the use of normalized versions of these random measures as priors in Bayesian nonparametric mixture models is considered. Their tractability allows the properties of both compound random measures and normalized compound random measures to be derived. In particular, we show how compound random measures can be constructed with gamma, $σ$-stable and generalized gamma process marginals. We also derive several forms of the Laplace exponent and characterize dependence through both the Lévy copula a…
Linear shrinkage estimators of a covariance matrix --- defined by a weighted average of the sample covariance matrix and a pre-specified shrinkage target matrix --- are popular when analysing high-throughput molecular data. However, their performance strongly relies on an appropriate choice of target matrix. This paper introduces a more flexible class of linear shrinkage estimators that can accommodate multiple shrinkage target matrices, directly accounting for the uncertainty regarding the target choice. This is done within a conjugate Bayesian framework, which is computationally efficient. U…
We propose a novel Dirichlet-based Pólya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Further, the continuity of the imposed D-P tree prior leads to a more favorable smoothing effect in copula estimation over classic frequentist metho…
Scalar-on-image regression aims to investigate changes in a scalar response of interest based on high-dimensional imaging data. We propose a novel Bayesian nonparametric scalar-on-image regression model that utilises the spatial coordinates of the voxels to group voxels with similar effects on the response to have a common coefficient. We employ the Potts-Gibbs random partition model as the prior for the random partition in which the partition process is spatially dependent, thereby encouraging groups representing spatially contiguous regions. In addition, Bayesian shrinkage priors are utilise…
Many recently developed Bayesian methods have focused on sparse signal detection. However, much less work has been done addressing the natural follow-up question: how to make valid inferences for the magnitude of those signals after selection. Ordinary Bayesian credible intervals suffer from selection bias, owing to the fact that the target of inference is chosen adaptively. Existing Bayesian approaches for correcting this bias produce credible intervals with poor frequentist properties, while existing frequentist approaches require sacrificing the benefits of shrinkage typical in Bayesian met…
Estimating a covariance matrix is an important task in applications where the number of variables is larger than the number of observations. Shrinkage approaches for estimating a high-dimensional covariance matrix are often employed to circumvent the limitations of the sample covariance matrix. A new family of nonparametric Stein-type shrinkage covariance estimators is proposed whose members are written as a convex linear combination of the sample covariance matrix and of a predefined invertible target matrix. Under the Frobenius norm criterion, the optimal shrinkage intensity that defines the…
In this paper we discuss Bayesian nonconvex penalization for sparse learning problems. We explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one-dimensional Lévy processes. We particularly study a family of continuous compound Poisson subordinators and a family of discrete compound Poisson subordinators. We exemplify four specific subordinators: Gamma, Poisson, negative binomial and squared Bessel subordinators. The Laplace exponents of the subordinators are Bernstein functions, so they can be used as sparsity-inducing nonconvex penalty functions.…
This paper describes a method to estimate a production frontier that satisfies the axioms of monotonicity and concavity in a non-parametric Bayesian setting. An inefficiency term that allows for significant departure from prior distributional assumptions is jointly estimated in a single stage with parametric prior assumptions. We introduce heteroscedasticity into the inefficiency terms by local hyperplane-specific shrinkage hyperparameters and impose monotonicity using bound-constrained local nonlinear regression. Our minimum-of-hyperplanes estimator imposes concavity. Our Monte Carlo simulati…
One way of defining probability distributions for circular variables (directions in two dimensions) is to radially project probability distributions, originally defined on $\mathbb{R}^2$, to the unit circle. Projected distributions have proved to be useful in the study of circular and directional data. Although any bivariate distribution can be used to produce a projected circular model, these distributions are typically parametric. In this article we consider a bivariate Pólya tree on $\mathbb{R}^2$ and project it to the unit circle to define a new Bayesian nonparametric model for circular da…
This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Polya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of the underlying distributions in a nonparametric way. The Bayesian perspective provides an inherently symmetric probability measure of conditional dependence or independence, a feature particularly advantageous in causal discovery and not employed in existing procedures of this type…
We develop a fully Bayesian hierarchical model for trend filtering, itself a new development in nonparametric, univariate regression. The framework more broadly applies to the generalized lasso, but focus is on Bayesian trend filtering. We compare two shrinkage priors, double exponential and generalized double Pareto. A simulation study, comparing Bayesian trend filtering to the original formulation and a number of other popular methods shows our method to improve estimation error while maintaining if not improving coverage probability. Two time series data sets demonstrate Bayesian trend filt…
The present paper investigates theoretical performance of various Bayesian wavelet shrinkage rules in a nonparametric regression model with i.i.d. errors which are not necessarily normally distributed. The main purpose is comparison of various Bayesian models in terms of their frequentist asymptotic optimality in Sobolev and Besov spaces. We establish a relationship between hyperparameters, verify that the majority of Bayesian models studied so far achieve theoretical optimality, state which Bayesian models cannot achieve optimal convergence rate and explain why it happens.
The edge partition model (EPM) is a fundamental Bayesian nonparametric model for extracting an overlapping structure from binary matrix. The EPM adopts a gamma process ($Γ$P) prior to automatically shrink the number of active atoms. However, we empirically found that the model shrinkage of the EPM does not typically work appropriately and leads to an overfitted solution. An analysis of the expectation of the EPM's intensity function suggested that the gamma priors for the EPM hyperparameters disturb the model shrinkage effect of the internal $Γ$P. In order to ensure that the model shrinkage ef…
Human microbiome studies use sequencing technologies to measure the abundance of bacterial species or Operational Taxonomic Units (OTUs) in samples of biological material. Typically the data are organized in contingency tables with OTU counts across heterogeneous biological samples. In the microbial ecology community, ordination methods are frequently used to investigate latent factors or clusters that capture and describe variations of OTU counts across biological samples. It remains important to evaluate how uncertainty in estimates of each biological sample's microbial distribution propagat…
The paper has two major themes. The first part of the paper establishes certain general results for infinite-dimensional optimization problems on Hilbert spaces. These results cover the classical representer theorem and many of its variants as special cases and offer a wider scope of applications. The second part of the paper then develops a systematic approach for learning the drift function of a stochastic differential equation by integrating the results of the first part with Bayesian hierarchical framework. Importantly, our Baysian approach incorporates low-cost sparse learning through pro…
We introduce a hierarchical nonparametric model for probability measures based on a multi-resolution transformation of probability distributions. The model allows a varying amount of shrinkage to be applied to data features of different scales and/or at different locations in the sample space, and the varying shrinkage level is locally adaptive to the empirical behavior of the data. Moreover, the model's hierarchical design---through a latent Markov tree structure---allows borrowing of information across locations and scales in setting the adaptive shrinkage level. Inference under the model pr…
Testing and characterizing the difference between two data samples is of fundamental interest in statistics. Existing methods such as Kolmogorov-Smirnov and Cramer-von-Mises tests do not scale well as the dimensionality increases and provides no easy way to characterize the difference should it exist. In this work, we propose a theoretical framework for inference that addresses these challenges in the form of a prior for Bayesian nonparametric analysis. The new prior is constructed based on a random-partition-and-assignment procedure similar to the one that defines the standard optional Pólya …
In the density estimation model, the question of adaptive inference using Pólya tree-type prior distributions is considered. A class of prior densities having a tree structure, called spike-and-slab Pólya trees, is introduced. For this class, two types of results are obtained: first, the Bayesian posterior distribution is shown to converge at the minimax rate for the supremum norm in an adaptive way, for any Hölder regularity of the true density between $0$ and $1$, thereby providing adaptive counterparts to the results for classical Pólya trees in Castillo (2017). Second, the question of unce…
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-parametric regression models. However, achieving sparsity simultaneously in the number of nonparametric components as well as in the variables within each nonparametric component poses a stiff computational challenge. In this article, we develop a novel Bayesian additive regressi…
Since Stein's 1956 seminal paper, shrinkage has played a fundamental role in both parametric and nonparametric inference. This article discusses minimaxity and adaptive minimaxity in nonparametric function estimation. Three interrelated problems, function estimation under global integrated squared error, estimation under pointwise squared error, and nonparametric confidence intervals, are considered. Shrinkage is pivotal in the development of both the minimax theory and the adaptation theory. While the three problems are closely connected and the minimax theories bear some similarities, the ad…
Symmetric binary matrices representing relations among entities are commonly collected in many areas. Our focus is on dynamically evolving binary relational matrices, with interest being in inference on the relationship structure and prediction. We propose a nonparametric Bayesian dynamic model, which reduces dimensionality in characterizing the binary matrix through a lower-dimensional latent space representation, with the latent coordinates evolving in continuous time via Gaussian processes. By using a logistic mapping function from the probability matrix space to the latent relational space…
Massachusetts Institute of Technology（麻省理工学院）；Pandora Media, Inc.（潘多拉媒体公司）；Stanford（斯坦福大学）
An important metric of users' satisfaction and engagement within on-line streaming services is the user session length, i.e. the amount of time they spend on a service continuously without interruption. Being able to predict this value directly benefits the recommendation and ad pacing contexts in music and video streaming services. Recent research has shown that predicting the exact amount of time spent is highly nontrivial due to many external factors for which a user can end a session, and the lack of predictive covariates. Most of the other related literature on duration based user engagem…
We investigate the posterior rate of convergence for wavelet shrinkage using a Bayesian approach in general Besov spaces. Instead of studying the Bayesian estimator related to a particular loss function, we focus on the posterior distribution itself from a nonparametric Bayesian asymptotics point of view and study its rate of convergence. We obtain the same rate as in \citet{abramovich04} where the authors studied the convergence of several Bayesian estimators.
We introduce a family of multiscale stick-breaking mixture models for Bayesian nonparametric density estimation. The Bayesian nonparametric literature is dominated by single scale methods, exception made for Pòlya trees and allied approaches. Our proposal is based on a mixture specification exploiting an infinitely-deep binary tree of random weights that grows according to a multiscale generalization of a large class of stick-breaking processes; this multiscale stick-breaking is paired with specific stochastic processes generating sequences of parameters that induce stochastically ordered kern…
Deakin University（迪肯大学）；University of Michigan（密歇根大学）；Nuance Communications（Nuance通信公司）
We present a Bayesian nonparametric framework for multilevel clustering which utilizes group-level context information to simultaneously discover low-dimensional structures of the group contents and partitions groups into clusters. Using the Dirichlet process as the building block, our model constructs a product base-measure with a nested structure to accommodate content and context observations at multiple levels. The proposed model possesses properties that link the nested Dirichlet processes (nDP) and the Dirichlet process mixture models (DPM) in an interesting way: integrating out all cont…
Pólya trees fix partitions and use random probabilities in order to construct random probability measures. With quantile pyramids we instead fix probabilities and use random partitions. For nonparametric Bayesian inference we use a prior which supports piecewise linear quantile functions, based on the need to work with a finite set of partitions, yet we show that the limiting version of the prior exists. We also discuss and investigate an alternative model based on the so-called substitute likelihood. Both approaches factorize in a convenient way leading to relatively straightforward analysis …
Discrete Bayesian nonparametric models whose expectation is a convex linear combination of a point mass at some point of the support and a diffuse probability distribution allow to incorporate strong prior information, while still being extremely flexible. Recent contributions in the statistical literature have successfully implemented such a modelling strategy in a variety of applications, including density estimation, nonparametric regression and model-based clustering. We provide a thorough study of a large class of nonparametric models we call inner spike and slab hNRMI models, which are o…
Recently, S. Arlot and R. Genuer have shown that a model of random forests outperforms its single-tree counterpart in the estimation of $α-$Hölder functions, $α\leq2$. This backs up the idea that ensembles of tree estimators are smoother estimators than single trees. On the other hand, most positive optimality results on Bayesian tree-based methods assume that $α\leq1$. Naturally, one wonders whether Bayesian counterparts of forest estimators are optimal on smoother classes, just like it has been observed for frequentist estimators for $α\leq 2$. We dwell on the problem of density estimation a…
