Random projections for Bayesian regression

This article deals with random projections applied as a data reduction technique for Bayesian regression analysis. We show sufficient conditions under which the entire $d$-dimensional distribution is approximately preserved under random projections by reducing the number of data points from $n$ to $k\in O(\operatorname{poly}(d/\varepsilon))$ in the case $n\gg d$. Under mild assumptions, we prove that evaluating a Gaussian likelihood function based on the projected data instead of the original data yields a $(1+O(\varepsilon))$-approximation in terms of the $\ell_2$ Wasserstein distance. Our main result shows that the posterior distribution of Bayesian linear regression is approximated up to a small error depending on only an $\varepsilon$-fraction of its defining parameters. This holds when using arbitrary Gaussian priors or the degenerate case of uniform distributions over $\mathbb{R}^d$ for $\beta$. Our empirical evaluations involve different simulated settings of Bayesian linear regression. Our experiments underline that the proposed method is able to recover the regression model up to small error while considerably reducing the total running time

Bayesian Regression of Piecewise Constant Functions

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 studies. We discuss the performance on synthetic and real-world examples. Many possible extensions will be discussed.Comment: 27 pages, 18 figures, 1 table, 3 algorithm

Robust Bayesian Regression with Synthetic Posterior

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 $\gamma$-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 to carry out robust Bayesian variable selection and estimation simultaneously. We develop an efficient posterior computation algorithm by adopting the Bayesian bootstrap within Gibbs sampling. The performance of the proposed method is illustrated through simulation studies and applications to famous datasets.Comment: 23 pages, 5 figure

On the Convergence of Bayesian Regression Models

We consider heteroscedastic nonparametric regression models, when both the mean function and variance function are unknown and to be estimated with nonparametric approaches. We derive convergence rates of posterior distributions for this model with different priors, including splines and Gaussian process priors. The results are based on the general ones on the rates of convergence of posterior distributions for independent, non-identically distributed observations, and are established for both of the cases with random covariates, and deterministic covariates. We also illustrate that the results can be achieved for all levels of regularity, which means they are adaptive

Bayesian Regression Markets

Machine learning tasks are vulnerable to the quality of data used as input. Yet, it is often challenging for firms to obtain adequate datasets, with them being naturally distributed amongst owners, that in practice, may be competitors in a downstream market and reluctant to share information. Focusing on supervised learning for regression tasks, we develop a \textit{regression market} to provide a monetary incentive for data sharing. Our proposed mechanism adopts a Bayesian framework, allowing us to consider a more general class of regression tasks. We present a thorough exploration of the market properties, and show that similar proposals in current literature expose the market agents to sizeable financial risks, which can be mitigated in our probabilistic setting.Comment: 46 pages, 11 figures, 2 table

Bayesian regression analysis.

Regression analysis is a statistical method used to relate a variable of interest, typically y (the dependent variable), to a set of independent variables, usually, X1, X2,...,Xn . The goal is to build a model that assists statisticians in describing, controlling, and predicting the dependent variable based on the independent variable(s). There are many types of regression analysis: Simple and Multiple Linear Regression, Nonlinear Regression, and Bayesian Regression Analysis to name a few. Here we will explore simple and multiple linear regression and Bayesian linear regression. For years, the most widely used method of regression analysis has been the Frequentist methods, or simple and multiple regression. However, with the advancements of computers and computing tools such as WinBUGS, Bayesian methods have become more widely accepted. With the use of WinBUGS, we can utilize a Markov Chain Monte Carlo (MCMC) method called Gibbs Sampling to simplify the increasingly difficult calculations. Given that Bayesian regression analysis is a relatively new method, it is not without faults. Many in the statistical community find that the use of Bayesian techniques is not a satisfactory method since the choice of the prior distribution is purely a guessing game and varies from statistician to statistician. In this thesis, an example is presented using both Frequentist and Bayesian methods and a comparison is made between the two. As computers become more advanced, the use of Bayesian regression analysis may become more widely accepted as the method of choice for regression analyses as it allows for the interpretation of a probability as a measure of degree of belief concerning actual data observed