Alternative posterior consistency results in nonparametric binary regression using Gaussian process priors

We establish consistency of posterior distribution when a Gaussian process prior is used as a prior distribution for the unknown binary regression function. Specifically, we take the work of Ghosal and Roy [2006. Posterior consistency of Gaussian process prior for nonparametric binary regression. Ann. Statist. 34, 2413–2429] as our starting point, and then weaken their assumptions on the smoothness of the Gaussian process kernel while retaining a stronger yet applicable condition about design points. Furthermore, we extend their results to multi-dimensional covariates under a weaker smoothness condition on the Gaussian process. Finally, we study the extent to which posterior consistency can be achieved under a general model where, when additional hyperparameters in the covariance function of a Gaussian process are involved.     

A note on Bayesian nonparametric regression function estimation

In this note the problem of nonparametric regression function estimation in a random design regression model with Gaussian errors is considered from the Bayesian perspective. It is assumed that the regression function belongs to a class of functions with a known degree of smoothness. A prior distribution on the given class can be induced by a prior on the coefficients in a series expansion of the regression function through an orthonormal system. The rate of convergence of the resulting posterior distribution is employed to provide a measure of the accuracy of the Bayesian estimation procedure defined by the posterior expected regression function. We show that the Bayes’ estimator achieves the optimal minimax rate of convergence under mean integrated squared error over the involved class of regression functions, thus being comparable to other popular frequentist regression estimators.     

Nonlinear orthogonal series estimates for random design regression

Let (X,Y) be a pair of random variables with supp(X)⊆[0,1] and EY²<∞. Let m be the corresponding regression function. Estimation of m from i.i.d. data is considered. The L₂ error with integration with respect to the design measure μ (i.e., the distribution of X) is used as an error criterion.Estimates are constructed by estimating the coefficients of an orthonormal expansion of the regression function. This orthonormal expansion is done with respect to a family of piecewise polynomials, which are orthonormal in L₂(μ_n), where μ_n denotes the empirical design measure.It is shown that the estimates are weakly and strongly consistent for every distribution of (X,Y). Furthermore, the estimates behave nearly as well as an ideal (but not applicable) estimate constructed by fitting a piecewise polynomial to the data, where the partition of the piecewise polynomial is chosen optimally for the underlying distribution. This implies e.g., that the estimates achieve up to a logarithmic factor the rate n^−2p/(2p+1), if the underlying regression function is piecewise p-smooth, although their definition depends neither on the smoothness nor on the location of the discontinuities of the regression function.     

Data reconciliation of nonnormal observations with nonlinear constraints

This paper presents a new method for the reconciliation of data described by arbitrary continuous probability distributions, with the focus on nonlinear constraints. The main idea, already applied to linear constraints in a previous paper, is to restrict the joint prior probability distribution of the observed variables with model constraints to get a joint posterior probability distribution. Because in general the posterior probability density function cannot be calculated analytically, it is shown that it has decisive advantages to sample from the posterior distribution by a Markov chain Monte Carlo (MCMC) method. From the resulting sample of observed and unobserved variables various characteristics of the posterior distribution can be estimated, such as the mean, the full covariance matrix, marginal posterior densities, as well as marginal moments, quantiles, and HPD intervals. The procedure is illustrated by examples from material flow analysis and chemical engineering.     

Convergence Analysis of MCMC Algorithms for Bayesian Multivariate Linear Regression with Non‐Gaussian Errors

When Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals. Combining this robust regression model with a default prior on the unknown parameters results in a highly intractable posterior density. Fortunately, there is a simple data augmentation (DA) algorithm and a corresponding Haar PX‐DA algorithm that can be used to explore this posterior. This paper provides conditions (on the mixing density) for geometric ergodicity of the Markov chains underlying these Markov chain Monte Carlo algorithms. Letting d denote the dimension of the response, the main result shows that the DA and Haar PX‐DA Markov chains are geometrically ergodic whenever the mixing density is generalized inverse Gaussian, log‐normal, inverted Gamma (with shape parameter larger than d /2) or Fréchet (with shape parameter larger than d /2). The results also apply to certain subsets of the Gamma, F and Weibull families.     

Adaptive Bayesian Procedures Using Random Series Priors

We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on Tecator data using functional regression models.     

带线性约束的多元线性回归模型参数估计

本文首先构造线性约束条件下的多元线性回归模型的样本似然函数,利用Lagrange法证明其合理性。其次,从似然函数的角度讨论线性约束条件对模型参数的影响,对由传统理论得出的参数估计作出贝叶斯与经验贝叶斯的改进。做贝叶斯改进时,将矩阵正态-Wishart分布作为模型参数和精度阵的联合共轭先验分布,结合构造的似然函数得出参数的后验分布,计算出参数的贝叶斯估计;做经验贝叶斯改进时,将样本分组,从方差的角度讨论由子样得出的参数估计对总样本的参数估计的影响,计算出经验贝叶斯估计。最后,利用Matlab软件生成的随机矩阵做模拟。结果表明,这两种改进后的参数估计均较由传统理论得出的参数估计更精确,拟合结果的误差比更小,可信度更高,在大数据的情况下,这种计算方法的速度更快。     

Stochastic matching pursuit for Bayesian variable selection

This article proposes a stochastic version of the matching pursuit algorithm for Bayesian variable selection in linear regression. In the Bayesian formulation, the prior distribution of each regression coefficient is assumed to be a mixture of a point mass at 0 and a normal distribution with zero mean and a large variance. The proposed stochastic matching pursuit algorithm is designed for sampling from the posterior distribution of the coefficients for the purpose of variable selection. The proposed algorithm can be considered a modification of the componentwise Gibbs sampler. In the componentwise Gibbs sampler, the variables are visited by a random or a systematic scan. In the stochastic matching pursuit algorithm, the variables that better align with the current residual vector are given higher probabilities of being visited. The proposed algorithm combines the efficiency of the matching pursuit algorithm and the Bayesian formulation with well defined prior distributions on coefficients. Several simulated examples of small n and large p are used to illustrate the algorithm. These examples show that the algorithm is efficient for screening and selecting variables.     

Multivariate Bayesian variable selection and prediction

The multivariate regression model is considered with p regressors. A latent vector with p binary entries serves to identify one of two types of regression coefficients: those close to 0 and those not. Specializing our general distributional setting to the linear model with Gaussian errors and using natural conjugate prior distributions, we derive the marginal posterior distribution of the binary latent vector. Fast algorithms aid its direct computation, and in high dimensions these are supplemented by a Markov chain Monte Carlo approach to sampling from the known posterior distribution. Problems with hundreds of regressor variables become quite feasible. We give a simple method of assigning the hyperparameters of the prior distribution. The posterior predictive distribution is derived and the approach illustrated on compositional analysis of data involving three sugars with 160 near infrared absorbances as regressors.     

Least absolute value regression: a special case of piecewise linear minimization

The Barrodale and Roberts algorithm for least absolute value (LAV) regression and the algorithm proposed by Bartels and Conn both have the advantage that they are often able to skip across points at which the conventional simplex-method algorithms for LAV regression would be required to carry out an (expensive) pivot operation.

We indicate here that this advantage holds in the Bartels-Conn approach for a wider class of problems: the minimization of piecewise linear functions. We show how LAV regression, restricted LAV regression, general linear programming and least maximum absolute value regression can all be easily expressed as piecewise linear minimization problems.     

A hierarchical Bayesian regression model for the uncertain functional constraint using screened scale mixtures of Gaussian distributions

This paper considers a hierarchical Bayesian analysis of regression models using a class of Gaussian scale mixtures. This class provides a robust alternative to the common use of the Gaussian distribution as a prior distribution in particular for estimating the regression function subject to uncertainty about the constraint. For this purpose, we use a family of rectangular screened multivariate scale mixtures of Gaussian distribution as a prior for the regression function, which is flexible enough to reflect the degrees of uncertainty about the functional constraint. Specifically, we propose a hierarchical Bayesian regression model for the constrained regression function with uncertainty on the basis of three stages of a prior hierarchy with Gaussian scale mixtures, referred to as a hierarchical screened scale mixture of Gaussian regression models (HSMGRM). We describe distributional properties of HSMGRM and an efficient Markov chain Monte Carlo algorithm for posterior inference, and apply the proposed model to real applications with constrained regression models subject to uncertainty.     

Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics

We find the distribution that has maximum entropy conditional on having specified values of its first r  L$L$-moments. This condition is equivalent to specifying the expected values of the order statistics of a sample of size r. The maximum-entropy distribution has a density-quantile function, the reciprocal of the derivative of the quantile function, that is a polynomial of degree r; the quantile function of the distribution can then be found by integration. This class of maximum-entropy distributions includes the uniform, exponential and logistic, and two new generalizations of the logistic distribution. It provides a new method of nonparametric fitting of a distribution to a data sample. We also derive maximum-entropy distributions subject to constraints on expected values of linear combinations of order statistics.     

LARS-type algorithm for group lasso

The least absolute shrinkage and selection operator (lasso) has been widely used in regression analysis. Based on the piecewise linear property of the solution path, least angle regression provides an efficient algorithm for computing the solution paths of lasso. Group lasso is an important generalization of lasso that can be applied to regression with grouped variables. However, the solution path of group lasso is not piecewise linear and hence cannot be obtained by least angle regression. By transforming the problem into a system of differential equations, we develop an algorithm for efficient computation of group lasso solution paths. Simulation studies are conducted for comparing the proposed algorithm to the best existing algorithm: the groupwise-majorization-descent algorithm.     

Variational Inference of Linear Regression with Nonzero Prior Means

In this article, we employ the variational Bayesian method to study the parameter estimation problems of linear regression model, wherein some regressors are of Gaussian distribution with nonzero prior means. We obtain an analytical expression of the posterior parameter distribution, and then propose an iterative algorithm for the model. Simulations are carried out to test the performance of the proposed algorithm, and the simulation results confirm both the effectiveness and the reliability of the proposed algorithm.     

Numerical Bayesian inference with arbitrary prior

The purpose of the paper, is to explain how recent advances in Markov Chain Monte Carlo integration can facilitate the routine Bayesian analysis of the linear model when the prior distribution is completely user dependent. The method is based on a Metropolis-Hastings algorithm with a Student-t source distribution that can generate posterior moments as well as marginal posterior densities for model parameters. The method is illustrated with numerical examples where the combination of prior and likelihood information leads to multimodal posteriors due to prior-likelihood conflicts, and to cases where prior information can be summarized by symmetric stable Paretian distributions.     

Exponential-Bound Property of Estimators and Variable Selection in Generalized Additive Models

In this article, utilizing a scale mixture of skew-normal distribution in which mixing random variable is assumed to follow a mixture model with varying weights for each observation, we introduce a generalization of skew-normal linear regression model with the aim to provide resistant results. This model, which also includes the skew-slash distribution in a particular case, allows us to accommodate and detect outlying observations under the skew-normal linear regression model. Inferences about the model are carried out through the empirical Bayes approach. The conditions for propriety of the posterior and for existence of posterior moments are given under the standard noninformative priors for regression and scale parameters as well as proper prior for skewness parameter. Then, for Bayesian inference, a Markov chain Monte Carlo method is described. Since posterior results depend on the prior hyperparameters, we estimate them adopting the empirical Bayes method as well as using a Monte Carlo EM algorithm. Furthermore, to identify possible outliers, we also apply the Bayes factor obtained through the generalized Savage-Dickey density ratio. Examining the proposed approach on simulated instance and real data, it is found to provide not only satisfactory parameter estimates rather allow identifying outliers favorably.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

A Matching Prior for the Shape Parameter of the Skew‐Normal Distribution

Abstract. This paper deals with the issue of performing a default Bayesian analysis on the shape parameter of the skew‐normal distribution. Our approach is based on a suitable pseudo‐likelihood function and a matching prior distribution for this parameter, when location (or regression) and scale parameters are unknown. This approach is important for both theoretical and practical reasons. From a theoretical perspective, it is shown that the proposed matching prior is proper thus inducing a proper posterior distribution for the shape parameter, also when the likelihood is monotone. From the practical perspective, the proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals.     

Markov Beta and Gamma Processes for Modelling Hazard Rates

This paper generalizes the discrete time independent increment beta process of Hjort (1990 ), for modelling discrete failure times, and also generalizes the independent gamma process for modelling piecewise constant hazard rates ( Walker and Mallick, 1997 ). The generalizations are from independent increment to Markov increment prior processes allowing the modelling of smoothness. We derive posterior distributions and undertake a full Bayesian analysis.     

NON‐PARAMETRIC BAYESIAN INFERENCE FOR INHOMOGENEOUS MARKOV POINT PROCESSES

With reference to a specific dataset, we consider how to perform a flexible non‐parametric Bayesian analysis of an inhomogeneous point pattern modelled by a Markov point process, with a location‐dependent first‐order term and pairwise interaction only. A priori we assume that the first‐order term is a shot noise process, and that the interaction function for a pair of points depends only on the distance between the two points and is a piecewise linear function modelled by a marked Poisson process. Simulation of the resulting posterior distribution using a Metropolis–Hastings algorithm in the ‘conventional’ way involves evaluating ratios of unknown normalizing constants. We avoid this problem by applying a recently introduced auxiliary variable technique. In the present setting, the auxiliary variable used is an example of a partially ordered Markov point process model.