13,765 research outputs found
Penalized Likelihood and Bayesian Function Selection in Regression Models
Challenging research in various fields has driven a wide range of
methodological advances in variable selection for regression models with
high-dimensional predictors. In comparison, selection of nonlinear functions in
models with additive predictors has been considered only more recently. Several
competing suggestions have been developed at about the same time and often do
not refer to each other. This article provides a state-of-the-art review on
function selection, focusing on penalized likelihood and Bayesian concepts,
relating various approaches to each other in a unified framework. In an
empirical comparison, also including boosting, we evaluate several methods
through applications to simulated and real data, thereby providing some
guidance on their performance in practice
Generalized structured additive regression based on Bayesian P-splines
Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM's and extensions to generalized structured additive regression based on one or two dimensional P-splines as the main building block. The approach extends previous work by Lang und Brezger (2003) for Gaussian responses. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. Our approach covers the most common univariate response distributions, e.g. the Binomial, Poisson or Gamma distribution, as well as multicategorical responses. For the first time, we present Bayesian semiparametric inference for the widely used multinomial logit models. As we will demonstrate through two applications on the forest health status of trees and a space-time analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX
Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review
The paper characterizes classes of functions for which deep learning can be
exponentially better than shallow learning. Deep convolutional networks are a
special case of these conditions, though weight sharing is not the main reason
for their exponential advantage
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