We study frequentist asymptotic properties of Bayesian procedures for high-dimensional Gaussian sparse regression when unknown nuisance parameters are involved. Nuisance parameters can be finite-, high-, or infinite-dimensional. A mixture of point masses at zero and continuous distributions is used for the prior distribution on sparse regression coefficients, and appropriate prior distributions are used for nuisance parameters. The optimal posterior contraction of sparse regression coefficients, hampered by the presence of nuisance parameters, is also examined and discussed. It is shown that the procedure yields strong model selection consistency. A Bernstein-von Mises-type theorem for sparse regression coefficients is also obtained for uncertainty quantification through credible sets with guaranteed frequentist coverage. Asymptotic properties of numerous examples are investigated using the theories developed in this study.
Prior distributions elicited for modelling the natural fluctuations or the uncertainty on parameters of Bayesian fishery population models, can be chosen among a vast range of statistical laws. Since the statistical framework is defined by observational processes, observational parameters enter into the estimation and must be considered random, similarly to parameters or states of interest like population levels or real catches. The former are thus perceived as nuisance parameters whose values are intrinsically linked to the considered experiment, which also require noninformative priors. In f…
Under a multinormal distribution with arbitrary unknown covariance matrix, the main purpose of this paper is to propose a framework to achieve the goal of reconciliation of Bayesian, frequentist and Fisherian paradigms for the problems of testing mean against restricted alternatives (closed convex cones). Combining Fisher's fiducial inference and Wald's decision theory via d-admissibility into an unified approach, the goal can then be achieved. To proceed, the tests constructed via the union-intersection principle are studied.
Northeastern University（东北大学）；Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室（MERL））
Learning data representations that capture task-related features, but are invariant to nuisance variations remains a key challenge in machine learning. We introduce an automated Bayesian inference framework, called AutoBayes, that explores different graphical models linking classifier, encoder, decoder, estimator and adversarial network blocks to optimize nuisance-invariant machine learning pipelines. AutoBayes also enables learning disentangled representations, where the latent variable is split into multiple pieces to impose various relationships with the nuisance variation and task labels. …
This paper is an early version. We propose to generalise the notion of "ignoring" a random process as well as the notions of informative and ignorable random processes in a very general setup and for different types of inference (Bayesian or frequentist), and for different purposes (estimation, prediction or testing). We then confront the definitions we propose to mentions or definitions of informative and ignorable processes found in the litterature. To that purpose, we provide a very general statistical framework for survey sampling in order to define precisely the notions of design and sele…
Bayesian Active Learning in the Presence of Nuisance Parameters
Department of Computer Science University of Manchester（计算机科学系曼彻斯特大学）；Department of Computer Science Aalto University（计算机科学系阿尔托大学）；Inserm Bordeaux Population Health, Vaccine Research Institute Université de Bordeaux Inria Bordeaux Sud-ouest（Inserm波尔多流行病与疫苗研究研究所波尔多大学Inria波尔多南西部）
We consider Bayesian inference in inverse regression problems where the objective is to infer about unobserved covariates from observed responses and covariates. We establish posterior consistency of such unobserved covariates in Bayesian inverse regression problemsunder appropriate priors in a leave-one-out cross-validation setup. We relate this to posterior consistency of inverse reference distributions (Bhattacharya (2013)) for assessing model adequacy. We illustrate our theory and methods with various examples of Bayesian inverse regression, along with adequate simulation experiments.
The asymmetric Laplace density (ALD) is used as a working likelihood for Bayesian quantile regression. Sriram et al.(2013) derived posterior consistency for Bayesian linear quantile regression based on the misspecified ALD. While their paper also argued for $\sqrt{n}-$consistency, Sriram and Ramamoorthi (2017) highlighted that the argument was only valid for a rate less than $\sqrt{n}$. So, the question of $\sqrt{n}-$rate has remained unaddressed, but is necessary to carry out meaningful Bayesian inference based on the ALD. In this paper, we derive results to obtain posterior consistency rates…
Many machine learning tools for regression are based on recursive partitioning of the covariate space into smaller regions, where the regression function can be estimated locally. Among these, regression trees and their ensembles have demonstrated impressive empirical performance. In this work, we shed light on the machinery behind Bayesian variants of these methods. In particular, we study Bayesian regression histograms, such as Bayesian dyadic trees, in the simple regression case with just one predictor. We focus on the reconstruction of regression surfaces that are piecewise constant, where…
This paper is focused on dimension-free PAC-Bayesian bounds, under weak polynomial moment assumptions, allowing for heavy tailed sample distributions. It covers the estimation of the mean of a vector or a matrix, with applications to least squares linear regression. Special efforts are devoted to the estimation of Gram matrices, due to their prominent role in high-dimension data analysis.
We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function. Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant. Then, we consider a class of priors for t…
In this paper we deal with the regression problem in a random design setting. We investigate asymptotic optimality under minimax point of view of various Bayesian rules based on warped wavelets and show that they nearly attain optimal minimax rates of convergence over the Besov smoothness class considered. Warped wavelets have been introduced recently, they offer very good computable and easy-to-implement properties while being well adapted to the statistical problem at hand. We particularly put emphasis on Bayesian rules leaning on small and large variance Gaussian priors and discuss their si…
We develop a unifying framework for Bayesian nonparametric regression to study the rates of contraction with respect to the integrated $L_2$-distance without assuming the regression function space to be uniformly bounded. The framework is very flexible and can be applied to a wide class of nonparametric prior models. Three non-trivial applications of the proposed framework are provided: The finite random series regression of an $α$-Hölder function, with adaptive rates of contraction up to a logarithmic factor; The un-modified block prior regression of an $α$-Sobolev function, with adaptive-and…
The paper deals with the non-parametric estimation in the regression with the multiplicative noise. Using the local polynomial fitting and the bayesian approach, we construct the minimax on isotropic Hölder class estimator. Next applying Lepski's method, we propose the estimator which is optimally adaptive over the collection of isotropic Hölder classes. To prove the optimality of the proposed procedure we establish, in particular, the exponential inequality for the deviation of locally bayesian estimator from the parameter to be estimated. These theoretical results are illustrated by simulati…
Since their inception in the 1980's, regression trees have been one of the more widely used non-parametric prediction methods. Tree-structured methods yield a histogram reconstruction of the regression surface, where the bins correspond to terminal nodes of recursive partitioning. Trees are powerful, yet susceptible to over-fitting. Strategies against overfitting have traditionally relied on pruning greedily grown trees. The Bayesian framework offers an alternative remedy against overfitting through priors. Roughly speaking, a good prior charges smaller trees where overfitting does not occur. …
In the problem of selecting variables in a multivariate linear regression model, we derive new Bayesian information criteria based on a prior mixing a smooth distribution and a delta distribution. Each of them can be interpreted as a fusion of the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Inheriting their asymptotic properties, our information criteria are consistent in variable selection in both the large-sample and the high-dimensional asymptotic frameworks. In numerical simulations, variable selection methods based on our information criteria choose th…
We review the Bayesian theory of semiparametric inference following Bickel and Kleijn (2012) and Kleijn and Knapik (2013). After an overview of efficiency in parametric and semiparametric estimation problems, we consider the Bernstein-von Mises theorem (see, e.g., Le Cam and Yang (1990)) and generalize it to (LAN) regular and (LAE) irregular semiparametric estimation problems. We formulate a version of the semiparametric Bernstein-von Mises theorem that does not depend on least-favourable submodels, thus bypassing the most restrictive condition in the presentation of Bickel and Kleijn (2012). …
The logistic regression model is the most popular model for analyzing binary data. In the absence of any prior information, an improper flat prior is often used for the regression coefficients in Bayesian logistic regression models. The resulting intractable posterior density can be explored by running Polson et al.'s (2013) data augmentation (DA) algorithm. In this paper, we establish that the Markov chain underlying Polson et al.'s (2013) DA algorithm is geometrically ergodic. Proving this theoretical result is practically important as it ensures the existence of central limit theorems (CLTs…
We investigate two empirical Bayes methods and a hierarchical Bayes method for adapting the scale of a Gaussian process prior in a nonparametric regression model. We show that all methods lead to a posterior contraction rate that adapts to the smoothness of the true regression function. Furthermore, we show that the corresponding credible sets cover the true regression function whenever this function satisfies a certain extrapolation condition. This condition depends on the specific method, but is implied by a condition of self-similarity. The latter condition is shown to be satisfied with pro…
We propose a new, two-step empirical Bayes-type of approach for neural networks. We show in context of the nonparametric regression model that the procedure (up to a logarithmic factor) provides optimal recovery of the underlying functional parameter of interest and provides Bayesian credible sets with frequentist coverage guarantees. The approach requires fitting the neural network only once, hence it is substantially faster than Bootstrapping type approaches. We demonstrate the applicability of our method over synthetic data, observing good estimation properties and reliable uncertainty quan…
For hypotheses of the type H_0:theta=theta_0 vs H_1:theta ne theta_0 we demonstrate the equivalence of a Bayesian hypothesis test using a Bayes factor and the corresponding classical test, for a large class of models, which are detailed in the paper. In particular, we show that the role of the prior and critical region for the Bayes factor test is only to specify the type I error. This is their only role since, as we show, the power function of the Bayes factor test coincides exactly with that of the classical test, once the type I error has been fixed. For more complex tests involving nuisanc…
Fitting models to data using Bayesian inference is quite common, but when each point in parameter space gives a curve, fitting the curve to a data set requires new nuisance parameters, which specify the metric embedding the one-dimensional curve into the higher-dimensional space occupied by the data. A generic formalism for curve fitting in the context of Bayesian inference is developed which shows how the aforementioned metric arises. The result is a natural generalization of previous works, and is compared to oft-used frequentist approaches and similar Bayesian techniques.
In this note we introduce linear regression with basis functions in order to apply Bayesian model selection. The goal is to incorporate Occam's razor as provided by Bayes analysis in order to automatically pick the model optimally able to explain the data without overfitting.
We consider three Bayesian penalized regression models and show that the respective deterministic scan Gibbs samplers are geometrically ergodic regardless of the dimension of the regression problem. We prove geometric ergodicity of the Gibbs samplers for the Bayesian fused lasso, the Bayesian group lasso, and the Bayesian sparse group lasso. Geometric ergodicity along with a moment condition results in the existence of a Markov chain central limit theorem for Monte Carlo averages and ensures reliable output analysis. Our results of geometric ergodicity allow us to also provide default starting…
In this paper, we discuss the method of Bayesian regression and its efficacy for predicting price variation of Bitcoin, a recently popularized virtual, cryptographic currency. Bayesian regression refers to utilizing empirical data as proxy to perform Bayesian inference. We utilize Bayesian regression for the so-called "latent source model". The Bayesian regression for "latent source model" was introduced and discussed by Chen, Nikolov and Shah (2013) and Bresler, Chen and Shah (2014) for the purpose of binary classification. They established theoretical as well as empirical efficacy of the met…
Bayesian Student-$t$ linear regression is a common robust alternative to the normal model, but its theoretical properties are not well understood. We aim to fill some gaps by providing analyses in two different asymptotic scenarios. The results allow to precisely characterize the trade-off between robustness and efficiency controlled through the degrees of freedom (at least asymptotically).
Posterior computation for high-dimensional data with many parameters can be challenging. This article focuses on a new method for approximating posterior distributions of a low- to moderate-dimensional parameter in the presence of a high-dimensional or otherwise computationally challenging nuisance parameter. The focus is on regression models and the key idea is to separate the likelihood into two components through a rotation. One component involves only the nuisance parameters, which can then be integrated out using a novel type of Gaussian approximation. We provide theory on approximation a…
Under model misspecification, the MLE generally converges to the pseudo-true parameter, the parameter corresponding to the distribution within the model that is closest to the distribution from which the data are sampled. In many problems, the pseudo-true parameter corresponds to a population parameter of interest, and so a misspecified model can provide consistent estimation for this parameter. Furthermore, the well-known sandwich variance formula of Huber(1967) provides an asymptotically accurate sampling distribution for the MLE, even under model misspecification. However, confidence interv…
We derive an exact and efficient Bayesian regression algorithm for piecewise constant functions of unknown segment number, boundary location, and levels. It works for any noise and segment level prior, e.g. Cauchy which can handle outliers. We derive simple but good estimates for the in-segment variance. We also propose a Bayesian regression curve as a better way of smoothing data without blurring boundaries. The Bayesian approach also allows straightforward determination of the evidence, break probabilities and error estimates, useful for model selection and significance and robustness studie…
In this paper, we consider the nonparametric random regression model $Y=f_1(X_1)+f_2(X_2)+ε$ and address the problem of estimating the function $f_1$. The term $f_2(X_2)$ is regarded as a nuisance term which can be considerably more complex than $f_1(X_1)$. Under minimal assumptions, we prove several nonasymptotic $L^2(\mathbb{P}^X)$-risk bounds for our estimators of $f_1$. Our approach is geometric and based on considerations in Hilbert spaces. It shows that the performance of our estimators is closely related to geometric quantities, such as minimal angles and Hilbert-Schmidt norms. Our resu…
Samsung AI Center（三星人工智能中心）；Rutgers University（罗格斯大学）
We propose a family of novel hierarchical Bayesian deep auto-encoder models capable of identifying disentangled factors of variability in data. While many recent attempts at factor disentanglement have focused on sophisticated learning objectives within the VAE framework, their choice of a standard normal as the latent factor prior is both suboptimal and detrimental to performance. Our key observation is that the disentangled latent variables responsible for major sources of variability, the relevant factors, can be more appropriately modeled using long-tail distributions. The typical Gaussian…
Bayesian penalized regression techniques, such as the Bayesian lasso and the Bayesian horseshoe estimator, have recently received a significant amount of attention in the statistics literature. However, software implementing state-of-the-art Bayesian penalized regression, outside of general purpose Markov chain Monte Carlo platforms such as STAN, is relatively rare. This paper introduces bayesreg, a new toolbox for fitting Bayesian penalized regression models with continuous shrinkage prior densities. The toolbox features Bayesian linear regression with Gaussian or heavy-tailed error models an…
This study aims to show the fundamental difference between logistic regression and Bayesian classifiers in the case of exponential and unexponential families of distributions, yielding the following findings. First, the logistic regression is a less general representation of a Bayesian classifier. Second, one should suppose distributions of classes for the correct specification of logistic regression equations. Third, in specific cases, there is no difference between predicted probabilities from correctly specified generative Bayesian classifier and discriminative logistic regression.
A two-stage approach is proposed to overcome the problem in quantile regression, where separately fitted curves for several quantiles may cross. The standard Bayesian quantile regression model is applied in the first stage, followed by a Gaussian process regression adjustment, which monotonizes the quantile function whilst borrowing strength from nearby quantiles. The two stage approach is computationally efficient, and more general than existing techniques. The method is shown to be competitive with alternative approaches via its performance in simulated examples.
In this paper we extend existing Bayesian methods for variable selection in Gaussian process regression, to select both the regression terms and the active covariates in the spatial correlation structure. We then use the estimated posterior probabilities to choose between relatively few modes through cross-validation, and consequently improve prediction.
Bayesian optimization (BO) is a model-based approach for gradient-free black-box function optimization. Typically, BO is powered by a Gaussian process (GP), whose algorithmic complexity is cubic in the number of evaluations. Hence, GP-based BO cannot leverage large amounts of past or related function evaluations, for example, to warm start the BO procedure. We develop a multiple adaptive Bayesian linear regression model as a scalable alternative whose complexity is linear in the number of observations. The multiple Bayesian linear regression models are coupled through a shared feedforward neur…
These notes aim at clarifying different strategies to perform linear regression from given dataset. Methods like the weighted and ordinary least squares, ridge regression or LASSO are proposed in the literature. The present article is my understanding of these methods which are, according to me, better unified in the Bayesian framework. The formulas to address linear regression with these methods are derived. The KIC for model selection is also derived in the end of the document.
Ensemble of regression trees have become popular statistical tools for the estimation of conditional mean given a set of predictors. However, quantile regression trees and their ensembles have not yet garnered much attention despite the increasing popularity of the linear quantile regression model. This work proposes a Bayesian quantile additive regression trees model that shows very good predictive performance illustrated using simulation studies and real data applications. Further extension to tackle binary classification problems is also considered.
Recently, a very attractive logistic regression inference method for exponential family Gibbs spatial point processes was introduced. We combined it with the technique of quadratic tangential variational approximation and derived a new Bayesian technique for analysing spatial point patterns. The technique is described in detail, and demonstrated on numerical examples.
In many longitudinal studies, the covariate and response are often intermittently observed at irregular, mismatched and subject-specific times. How to deal with such data when covariate and response are observed asynchronously is an often raised problem. Bayesian Additive Regression Trees(BART) is a Bayesian non-Parametric approach which has been shown to be competitive with the best modern predictive methods such as random forest and boosted decision trees. The sum of trees structure combined with a Bayesian inferential framework provide a accurate and robust statistic method. BART variant so…
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 binary random variables that are mutually independent given their covariate-dependent stick success probabilities, which are parameterized by the regression coefficients of their corresponding categories. The paSB construction allows transforming an arbitrary cross-entropy-loss binary…
Being in the era of big data, modeling and prediction of count data have become significantly important in many fields including health, finance, social, etc. Although linear Poisson regression has been widely used to model count and rate data, it might not be always suitable as it cannot capture some inherent variability within complex data. In this study, we introduce a probabilistically driven nonlinear Poisson regression model with Bayesian artificial neural networks (ANN) to model count or rate data. This new nonlinear Poisson regression model developed with Bayesian ANN provides higher p…
This article proposes Multinomial Probit Bayesian Additive Regression Trees (MPBART) as a multinomial probit extension of BART - Bayesian Additive Regression Trees (Chipman et al (2010)). MPBART is flexible to allow inclusion of predictors that describe the observed units as well as the available choice alternatives. Through two simulation studies and four real data examples, we show that MPBART exhibits very good predictive performance in comparison to other discrete choice and multiclass classification methods. To implement MPBART, we have developed an R package mpbart available freely from …
In the linear regression model with possibly autoregressive errors, we propose a family of nonparametric tests for regression under a nuisance autoregression. The tests avoid the estimation of nuisance parameters, in contrast to the tests proposed in the literature.
Regression for spatially dependent outcomes poses many challenges, for inference and for computation. Non-spatial models and traditional spatial mixed-effects models each have their advantages and disadvantages, making it difficult for practitioners to determine how to carry out a spatial regression analysis. We discuss the data-generating mechanisms implicitly assumed by various popular spatial regression models, and discuss the implications of these assumptions. We propose Bayesian spatial filtering as an approximate middle way between non-spatial models and traditional spatial mixed models.…
We propose Bayesian extensions of two nonparametric regression methods which are kernel and mutual $k$-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 shown that both the proposed methods asymptotically converge to kernel and mutual $k$-nearest neighbor regression methods, respectively. The simulation results show that the proposed methods can select proper hyperparameters and are better than or comparable to the former methods for…
Although linear regression models are fundamental tools in statistical science, the estimation results can be sensitive to outliers. While several robust methods have been proposed in frequentist frameworks, statistical inference is not necessarily straightforward. We here propose a Bayesian approach to robust inference on linear regression models using synthetic posterior distributions based on $γ$-divergence, which enables us to naturally assess the uncertainty of the estimation through the posterior distribution. We also consider the use of shrinkage priors for the regression coefficients t…
Bayesian methods are increasingly applied in these days in the theory and practice of statistics. Any Bayesian inference depends on a likelihood and a prior. Ideally one would like to elicit a prior from related sources of information or past data. However, in its absence, Bayesian methods need to rely on some "objective" or "default" priors, and the resulting posterior inference can still be quite valuable. Not surprisingly, over the years, the catalog of objective priors also has become prohibitively large, and one has to set some specific criteria for the selection of such priors. Our aim i…
As an automatic method of determining model complexity using the training data alone, Bayesian linear regression provides us a principled way to select hyperparameters. But one often needs approximation inference if distribution assumption is beyond Gaussian distribution. In this paper, we propose a Bayesian linear regression model with Student-t assumptions (BLRS), which can be inferred exactly. In this framework, both conjugate prior and expectation maximization (EM) algorithm are generalized. Meanwhile, we prove that the maximum likelihood solution is equivalent to the standard Bayesian lin…
This paper builds on recent research that focuses on regression modeling of continuous bounded data, such as proportions measured on a continuous scale. Specifically, it deals with beta regression models with mixed effects from a Bayesian approach. We use a suitable parameterization of the beta law in terms of its mean and a precision parameter, and allow both parameters to be modeled through regression structures that may involve fixed and random effects. Specification of prior distributions is discussed, computational implementation via Gibbs sampling is provided, and illustrative examples a…
COVID-19 incidence is analyzed at the provinces of some Spanish Communities during the period February-October, 2020. Two infinite-dimensional regression approaches are tested. The first one is implemented in the regression framework introduced in Ruiz-Medina, Miranda and Espejo (2019). Specifically, a bayesian framework is adopted in the estimation of the pure point spectrum of the temporal autocorrelation operator, characterizing the second-order structure of a surface sequence. The second approach is formulated in the context of spatial curve regression. A nonparametric estimator of the spe…
