To solve a machine learning problem, one typically needs to perform data preprocessing, modeling, and hyperparameter tuning, which is known as model selection and hyperparameter optimization.The goal of automated machine learning (AutoML) is to design methods that can automatically perform model selection and hyperparameter optimization without human interventions for a given dataset. In this paper, we propose a meta-learning method that can search for a high-performance machine learning pipeline from the predefined set of candidate pipelines for supervised classification datasets in an efficient way by leveraging meta-data collected from previous experiments. More specifically, our method combines an adaptive Bayesian regression model with a neural network basis function and the acquisition function from Bayesian optimization. The adaptive Bayesian regression model is able to capture knowledge from previous meta-data and thus make predictions of the performances of machine learning pipelines on a new dataset. The acquisition function is then used to guide the search of possible pipelines based on the predictions.The experiments demonstrate that our approach can quickly identify high-performance pipelines for a range of test datasets and outperforms the baseline methods.
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…
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…
SoftServe, Inc.（SoftServe 公司）；Ivan Franko National University of Lviv（利沃夫伊万·弗兰科国立大学）
In this work, we study the use of logistic regression in manufacturing failures detection. As a data set for the analysis, we used the data from Kaggle competition Bosch Production Line Performance. We considered the use of machine learning, linear and Bayesian models. For machine learning approach, we analyzed XGBoost tree based classifier to obtain high scored classification. Using the generalized linear model for logistic regression makes it possible to analyze the influence of the factors under study. The Bayesian approach for logistic regression gives the statistical distribution for the …
We present a new package in R implementing Bayesian additive regression trees (BART). The package introduces many new features for data analysis using BART such as variable selection, interaction detection, model diagnostic plots, incorporation of missing data and the ability to save trees for future prediction. It is significantly faster than the current R implementation, parallelized, and capable of handling both large sample sizes and high-dimensional data.
Linear regression is an important tool across many fields that work with sensitive human-sourced data. Significant prior work has focused on producing differentially private point estimates, which provide a privacy guarantee to individuals while still allowing modelers to draw insights from data by estimating regression coefficients. We investigate the problem of Bayesian linear regression, with the goal of computing posterior distributions that correctly quantify uncertainty given privately released statistics. We show that a naive approach that ignores the noise injected by the privacy mecha…
University of Pennsylvania（宾夕法尼亚大学）；Medical College of Wisconsin（威斯康星医学院）；Arizona State University（亚利桑那州立大学）
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 normal errors. This strong parametric assumption can lead to misleading inference and uncertainty quantification. In this paper, we use the classic Dirichlet process mixture (DPM) mechanism to nonparametrically model the error distribution. A key strength of BART is that default pri…
Neural Linear Models (NLM) are deep Bayesian models that produce predictive uncertainties by learning features from the data and then performing Bayesian linear regression over these features. Despite their popularity, few works have focused on methodically evaluating the predictive uncertainties of these models. In this work, we demonstrate that traditional training procedures for NLMs drastically underestimate uncertainty on out-of-distribution inputs, and that they therefore cannot be naively deployed in risk-sensitive applications. We identify the underlying reasons for this behavior and p…
Aalto University（阿尔托大学）；University of Helsinki（赫尔辛基大学）；Institute for Molecular Medicine Finland (FIMM)（芬兰分子医学研究所（FIMM））；University of Oulu（奥卢大学）；Oulu University Hospital（奥卢大学医院）；University of Eastern Finland（东芬兰大学）；Imperial College London（帝国理工学院）；National Institute for Health and Welfare（国家健康与福利研究所）；University of Bristol（布里斯托大学）
We consider the prediction of weak effects in a multiple-output regression setup, when covariates are expected to explain a small amount, less than $\approx 1%$, of the variance of the target variables. To facilitate the prediction of the weak effects, we constrain our model structure by introducing a novel Bayesian approach of sharing information between the regression model and the noise model. Further reduction of the effective number of parameters is achieved by introducing an infinite shrinkage prior and group sparsity in the context of the Bayesian reduced rank regression, and using the …
A simple approach to obtaining uncertainty-aware neural networks for regression is to do Bayesian linear regression (BLR) on the representation from the last hidden layer. Recent work [Riquelme et al., 2018, Azizzadenesheli et al., 2018] indicates that the method is promising, though it has been limited to homoscedastic noise. In this paper, we propose a novel variation that enables the method to flexibly model heteroscedastic noise. The method is benchmarked against two prominent alternative methods on a set of standard datasets, and finally evaluated as an uncertainty-aware model in model-ba…
Tokyo Institute of Technology（东京工业大学）；Hiroshima University（广岛大学）
Cross-validation (CV) is a technique for evaluating the ability of statistical models/learning systems based on a given data set. Despite its wide applicability, the rather heavy computational cost can prevent its use as the system size grows. To resolve this difficulty in the case of Bayesian linear regression, we develop a formula for evaluating the leave-one-out CV error approximately without actually performing CV. The usefulness of the developed formula is tested by statistical mechanical analysis for a synthetic model. This is confirmed by application to a real-world supernova data set a…
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…
Bayesian additive regression trees have seen increased interest in recent years due to their ability to combine machine learning techniques with principled uncertainty quantification. The Bayesian backfitting algorithm used to fit BART models, however, limits their application to a small class of models for which conditional conjugacy exists. In this article, we greatly expand the domain of applicability of BART to arbitrary \emph{generalized BART} models by introducing a very simple, tuning-parameter-free, reversible jump Markov chain Monte Carlo algorithm. Our algorithm requires only that th…
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.
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 …
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…
Online Passive-Aggressive (PA) learning is a class of online margin-based algorithms suitable for a wide range of real-time prediction tasks, including classification and regression. PA algorithms are formulated in terms of deterministic point-estimation problems governed by a set of user-defined hyperparameters: the approach fails to capture model/prediction uncertainty and makes their performance highly sensitive to hyperparameter configurations. In this paper, we introduce a novel PA learning framework for regression that overcomes the above limitations. We contribute a Bayesian state-space…
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…
Robust Bayesian models are appealing alternatives to standard models, providing protection from data that contains outliers or other departures from the model assumptions. Historically, robust models were mostly developed on a case-by-case basis; examples include robust linear regression, robust mixture models, and bursty topic models. In this paper we develop a general approach to robust Bayesian modeling. We show how to turn an existing Bayesian model into a robust model, and then develop a generic strategy for computing with it. We use our method to study robust variants of several models, …
We consider the bridge linear regression modeling, which can produce a sparse or non-sparse model. A crucial point in the model building process is the selection of adjusted parameters including a regularization parameter and a tuning parameter in bridge regression models. The choice of the adjusted parameters can be viewed as a model selection and evaluation problem. We propose a model selection criterion for evaluating bridge regression models in terms of Bayesian approach. This selection criterion enables us to select the adjusted parameters objectively. We investigate the effectiveness of …
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.
In linear regression models, fusion of coefficients is used to identify predictors having similar relationships with a response. This is called variable fusion. This paper presents a novel variable fusion method in terms of Bayesian linear regression models. We focus on hierarchical Bayesian models based on a spike-and-slab prior approach. A spike-and-slab prior is tailored to perform variable fusion. To obtain estimates of the parameters, we develop a Gibbs sampler for the parameters. Simulation studies and a real data analysis show that our proposed method achieves better performance than pr…
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…
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.
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.
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.
In this rejoinder we summarize the comments, questions and remarks on the paper "A novel algorithmic approach to Bayesian Logic Regression" from the discussants. We then respond to those comments, questions and remarks, provide several extensions of the original model and give a tutorial on our R-package EMJMCMC (http://aliaksah.github.io/EMJMCMC2016/)
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…
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.
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.
Bayesian additive regression tree (BART) models have seen increased attention in recent years as a general-purpose nonparametric modeling technique. BART combines the flexibility of modern machine learning techniques with the principled uncertainty quantification of Bayesian inference, and it has been shown to be uniquely appropriate for addressing the high-noise problems that occur commonly in many areas of science, including medicine and the social sciences. This paper introduces the SoftBart package for fitting the Soft BART algorithm of Linero and Yang (2018). In addition to improving upon…
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…
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…
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…
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…
Leiden University（莱顿大学）；CWI（荷兰国家数学与计算机科学研究所）；University of Oxford（牛津大学）；University of Victoria（维多利亚大学）
We study generalized Bayesian inference under misspecification, i.e. when the model is 'wrong but useful'. Generalized Bayes equips the likelihood with a learning rate $η$. We show that for generalized linear models (GLMs), $η$-generalized Bayes concentrates around the best approximation of the truth within the model for specific $η\neq 1$, even under severely misspecified noise, as long as the tails of the true distribution are exponential. We derive MCMC samplers for generalized Bayesian lasso and logistic regression and give examples of both simulated and real-world data in which generalize…
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.…
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).
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…
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 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…
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.
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…
This book chapter introduces regression approaches and regression adjustment for Approximate Bayesian Computation (ABC). Regression adjustment adjusts parameter values after rejection sampling in order to account for the imperfect match between simulations and observations. Imperfect match between simulations and observations can be more pronounced when there are many summary statistics, a phenomenon coined as the curse of dimensionality. Because of this imperfect match, credibility intervals obtained with regression approaches can be inflated compared to true credibility intervals. The chapte…
Like mean, quantile and variance, mode is also an important measure of central tendency and data summary. Many practical questions often focus on "Which element (gene or file or signal) occurs most often or is the most typical among all elements in a network?". In such cases mode regression provides a convenient summary of how the regressors affect the conditional mode and is totally different from other regression models based on conditional mean or conditional quantile or conditional variance. Some inference methods have been used for mode regression but none of them from the Bayesian perspe…
This paper proposes new linear regression models to deal with overdispersed binomial datasets. These new models, called tilted beta binomial regression models, are defined from the tilted beta binomial distribution, proposed assuming that the parameter of the binomial distribution follows a tilted beta distribution. As a particular case of this regression models, we propose the beta rectangular binomial regression models, defined from the binomial distribution assuming that their parameters follow a beta rectangular distribution. These new linear regression models, defined assuming that the pa…
We consider monotonic, multiple regression for a set of contiguous regions (lattice data). The regression functions permissibly vary between regions and exhibit geographical structure. We develop new Bayesian non-parametric methodology which allows for both continuous and discontinuous functional shapes and which are estimated using marked point processes and reversible jump Markov Chain Monte Carlo techniques. Geographical dependency is incorporated by a flexible prior distribution; the parametrisation allows the dependency to vary with functional level. The approach is tuned using Bayesian g…
We develop a fully Bayesian, logistic tracking algorithm with the purpose of providing classification results that are unbiased when applied uniformly to individuals with differing sensitive variable values. Here, we consider bias in the form of differences in false prediction rates between the different sensitive variable groups. Given that the method is fully Bayesian, it is well suited for situations where group parameters or logistic regression coefficients are dynamic quantities. We illustrate our method, in comparison to others, on both simulated datasets and the well-known ProPublica CO…
