### Bayesian Approximate Kernel Regression with Variable Selection

Nonlinear kernel regression models are often used in statistics and machine learning because they are more accurate than linear models. Variable selection for kernel regression models is a challenge partly because, unlike the linear regression setting, there is no clear concept of an effect size for regression coefficients. In this paper, we propose a novel framework that provides an effect size analog of each explanatory variable for Bayesian kernel regression models when the kernel is shift-invariant --- for example, the Gaussian kernel. We use function analytic properties of shift-invariant reproducing kernel Hilbert spaces (RKHS) to define a linear vector space that: (i) captures nonlinear structure, and (ii) can be projected onto the original explanatory variables. The projection onto the original explanatory variables serves as an analog of effect sizes. The specific function analytic property we use is that shift-invariant kernel functions can be approximated via random Fourier bases. Based on the random Fourier expansion we propose a computationally efficient class of Bayesian approximate kernel regression (BAKR) models for both nonlinear regression and binary classification for which one can compute an analog of effect sizes. We illustrate the utility of BAKR by examining two important problems in statistical genetics: genomic selection (i.e. phenotypic prediction) and association mapping (i.e. inference of significant variants or loci). State-of-the-art methods for genomic selection and association mapping are based on kernel regression and linear models, respectively. BAKR is the first method that is competitive in both settings.

### Predictor Variable Prioritization in Nonlinear Models: A Genetic Association Case Study

The central aim in this paper is to address variable selection questions in nonlinear and nonparametric regression. Motivated by statistical genetics, where nonlinear interactions are of particular interest, we introduce a novel, interpretable, and computationally efficient way to summarize the relative importance of predictor variables. Methodologically, we develop the "RelATive cEntrality" (RATE) measure to prioritize candidate genetic variants that are not just marginally important, but whose associations also stem from significant covarying relationships with other variants in the data. We illustrate RATE through Bayesian Gaussian process regression, but the methodological innovations apply to other nonlinear methods. It is known that nonlinear models often exhibit greater predictive accuracy than linear models, particularly for phenotypes generated by complex genetic architectures. With detailed simulations and an Arabidopsis thaliana QTL mapping study, we show that applying RATE enables an explanation for this improved performance.

### Fast Gaussian Process Regression for Big Data

Gaussian Processes are widely used for regression tasks. A known limitation in the application of Gaussian Processes to regression tasks is that the computation of the solution requires performing a matrix inversion. The solution also requires the storage of a large matrix in memory. These factors restrict the application of Gaussian Process regression to small and moderate size data sets. We present an algorithm that combines estimates from models developed using subsets of the data obtained in a manner similar to the bootstrap. The sample size is a critical parameter for this algorithm. Guidelines for reasonable choices of algorithm parameters, based on detailed experimental study, are provided. Various techniques have been proposed to scale Gaussian Processes to large scale regression tasks. The most appropriate choice depends on the problem context. The proposed method is most appropriate for problems where an additive model works well and the response depends on a small number of features. The minimax rate of convergence for such problems is attractive and we can build effective models with a small subset of the data. The Stochastic Variational Gaussian Process and the Sparse Gaussian Process are also appropriate choices for such problems. These methods pick a subset of data based on theoretical considerations. The proposed algorithm uses bagging and random sampling. Results from experiments conducted as part of this study indicate that the algorithm presented in this work can be as effective as these methods. Model stacking can be used to combine the model developed with the proposed method with models from other methods for large scale regression such as Gradient Boosted Trees. This can yield performance gains.

Sparse generalized additive models (GAMs) are an extension of sparse generalized linear models which allow a model's prediction to vary non-linearly with an input variable. This enables the data analyst build more accurate models, especially when the linearity assumption is known to be a poor approximation of reality. Motivated by reluctant interaction modeling (Yu et al. 2019), we propose a multi-stage algorithm, called $\textit{reluctant additive modeling (RAM)}$, that can fit sparse generalized additive models at scale. It is guided by the principle that, if all else is equal, one should prefer a linear feature over a non-linear feature. Unlike existing methods for sparse GAMs, RAM can be extended easily to binary, count and survival data. We demonstrate the method's effectiveness on real and simulated examples.