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.Comment: 22 pages, 3 figures, 3 tables; theory added; new simulations presented; references adde

Entropy inference and the James-Stein estimator, with application to nonlinear gene association networks

We present a procedure for effective estimation of entropy and mutual information from small-sample data, and apply it to the problem of inferring high-dimensional gene association networks. Specifically, we develop a James-Stein-type shrinkage estimator, resulting in a procedure that is highly efficient statistically as well as computationally. Despite its simplicity, we show that it outperforms eight other entropy estimation procedures across a diverse range of sampling scenarios and data-generating models, even in cases of severe undersampling. We illustrate the approach by analyzing E. coli gene expression data and computing an entropy-based gene-association network from gene expression data. A computer program is available that implements the proposed shrinkage estimator.Comment: 18 pages, 3 figures, 1 tabl

Nonparametric Bayes dynamic modeling of relational data

Symmetric binary matrices representing relations among entities are commonly collected in many areas. Our focus is on dynamically evolving binary relational matrices, with interest being in inference on the relationship structure and prediction. We propose a nonparametric Bayesian dynamic model, which reduces dimensionality in characterizing the binary matrix through a lower-dimensional latent space representation, with the latent coordinates evolving in continuous time via Gaussian processes. By using a logistic mapping function from the probability matrix space to the latent relational space, we obtain a flexible and computational tractable formulation. Employing P\`olya-Gamma data augmentation, an efficient Gibbs sampler is developed for posterior computation, with the dimension of the latent space automatically inferred. We provide some theoretical results on flexibility of the model, and illustrate performance via simulation experiments. We also consider an application to co-movements in world financial markets

Implicit Copulas from Bayesian Regularized Regression Smoothers

We show how to extract the implicit copula of a response vector from a Bayesian regularized regression smoother with Gaussian disturbances. The copula can be used to compare smoothers that employ different shrinkage priors and function bases. We illustrate with three popular choices of shrinkage priors --- a pairwise prior, the horseshoe prior and a g prior augmented with a point mass as employed for Bayesian variable selection --- and both univariate and multivariate function bases. The implicit copulas are high-dimensional, have flexible dependence structures that are far from that of a Gaussian copula, and are unavailable in closed form. However, we show how they can be evaluated by first constructing a Gaussian copula conditional on the regularization parameters, and then integrating over these. Combined with non-parametric margins the regularized smoothers can be used to model the distribution of non-Gaussian univariate responses conditional on the covariates. Efficient Markov chain Monte Carlo schemes for evaluating the copula are given for this case. Using both simulated and real data, we show how such copula smoothing models can improve the quality of resulting function estimates and predictive distributions

Scale effects on genomic modelling and prediction

In dieser Arbeit wird eine neue Methode für den skalenunabhängigen Vergleich von LD-Strukturen in unterschiedlichen genomischen Regionen vorgeschlagen. Verschiedene Aspekte durch Skalen verursachter Probleme – von der Präzision der Schätzung der Marke-reffekte bis zur Genauigkeit der Vorhersage für neue Individuen - wurden untersucht. Darüber hinaus, basierend auf den Leistungsvergleichen von unterschiedlichen statistischen Methoden, wurden Empfehlungen für die Verwendungen der untersuchten Methoden gege-ben.In dieser Arbeit wird eine neue Methode für den skalenunabhängigen Vergleich von LD-Strukturen in unterschiedlichen genomischen Regionen vorgeschlagen. Verschiedene Aspekte durch Skalen verursachter Probleme – von der Präzision der Schätzung der Marke-reffekte bis zur Genauigkeit der Vorhersage für neue Individuen - wurden untersucht. Darüber hinaus, basierend auf den Leistungsvergleichen von unterschiedlichen statistischen Methoden, wurden Empfehlungen für die Verwendungen der untersuchten Methoden gegebenIn this thesis, a novel method for scale corrected comparisons of LD structure in different genomic regions is suggested. Several aspects of scale-caused problems – from precision of marker effect estimates to accuracy of predictions for new individuals - are investigated. Furthermore, based on a comparison of the performance of different approaches, recommendations on the application of examined methods are given

A Dirichlet process mixture model for survival outcome data: assessing nationwide kidney transplant centers

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/110837/1/sim6438.pd