In principle, the Bayesian approach to model selection is straightforward. Prior probability distributions are used to describe the uncertainty surrounding all unknowns. After observing the data, the posterior distribution provides a coherent post data summary of the remaining uncertainty which is relevant for model selection. However, the practical implementation of this approach often requires carefully tailored priors and novel posterior calculation methods. In this article, we illustrate some of the fundamental practical issues that arise for two different model selection problems: the variable selection problem for the linear model and the CART model selection problem.

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A Bayesian method is presented for the nonparametric modelling of univariate and multivariate non-Gaussian response data. Data adaptive multivariate smoothing splines are employed where the number and location of the knot points are treated as random. The posterior model space is explored using a reversible jump Markov chain Monte Carlo sampler. Computational difficulties are partly alleviated by introducing a residual effect in the model that leaves many of the posterior distributions of the model parameters in standard form. The use of the latent residual effect provides a convenient vehicle for modelling correlation in multivariate response data and as such our method can be seen to generalize the seemingly unrelated regression model (Zellner, 1962) to non-Gaussian data. KEYWORDS: Bayesian nonlinear regression; multivariate splines; piecewise linear; local linear regression; SUR; multivariate nonlinear regression; generalised nonlinear regression. 1 1 Introduction R...

Abstract: We use rescaled Gaussian processes as prior models for functional parameters in nonparametric statistical models. We show how the rate of contraction of the posterior distributions depends on the scaling factor. In particular, we exhibit rescaled Gaussian process priors yielding posteriors that contract around the true parameter at optimal convergence rates. To derive our results we establish bounds on small deviation probabilities for smooth stationary Gaussian processes.

Statistical Analysis of Cell Motion by Edward Luke Ionides Doctor of Philosophy in Statistics University of California, Berkeley Professor David R. Brillinger, Chair Certain biological experiments investigating cell motion result in time lapse video microscopy data which may be modeled using stochastic di#erential equations. These models suggest statistics for quantifying experimental results and testing relevant hypotheses, and carry implications for the qualitative behavior of cells and for underlying biophysical mechanisms. A state space model formulation is used to link models proposed for cell velocity to observed data. Sequential Monte Carlo methods enable parameter estimation and model assessment for a range of applicable models. One particular experimental situation, involving the e#ect of an electric field on cell behavior, is considered in detail.

This paper describes and compares some approaches to this problem in the context of nonparametric regression.

This article deals with the analysis of a hierarchical semiparametric model for dynamic binary longitudinal responses. The main complicating components of the model are an unknown covariate function and serial correlation in the errors. Existing estimation methods for models with these features are of O(N3),whereNis the total number of observations in the sample. Therefore, nonparametric estimation is largely infeasible when the sample size is large, as in typical in the longitudinal setting. Here we propose a new O(N) Markov chain Monte Carlo based algorithm for estimation of the nonparametric function when the errors are correlated, thus contributing to the growing literature on semiparametric and nonparametric mixed-effects models for binary data. In addition, we address the problem of model choice to enable the formal comparison of our semiparametric model with competing parametric and semiparametric specifications. The performance of the methods is illustrated with detailed studies involving simulated and real data.

We propose a semiparametric Bayesian method for handling measurement error in nutritional epidemiological data. Our goal is to estimate nonparametrically the form of association between a disease and exposure variable while the true values of the exposure are never observed. Motivated by nutritional epidemiological data we consider the setting where a surrogate covariate is recorded in the primary data, and a calibration data set contains information on the surrogate variable and repeated measurements of an unbiased instrumental variable of the true exposure. We develop a flexible Bayesian method where not only is the relationship between the disease and exposure variable treated semiparametrically, but also the relationship between the surrogate and the true exposure is modeled semiparametrically. The two nonparametric functions are modeled simultaneously via B-splines. In addition, we model the distribution of the exposure variable as a Dirichlet process mixture of normal distributions, thus making its modeling essentially nonparametric and placing this work into the context of functional measurement error modeling. We apply our method to the NIH-AARP Diet and Health Study and examine its performance in a simulation study.

www.elsevier.com/locate/stamet Nonparametric binary regression using a Gaussian process prior

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