1,266 research outputs found
Locally Adaptive Bayesian P-Splines with a Normal-Exponential-Gamma Prior
The necessity to replace smoothing approaches with a global amount of smoothing arises in a variety of situations such as effects with highly varying curvature or effects with discontinuities. We present an implementation of locally adaptive spline smoothing using a class of heavy-tailed shrinkage priors. These priors utilize scale mixtures of normals with locally varying exponential-gamma distributed variances for the differences of the P-spline coefficients. A fully Bayesian hierarchical structure is derived with inference about the posterior being based on Markov Chain Monte Carlo techniques. Three increasingly flexible and automatic approaches are introduced to estimate the spatially varying structure of the variances. In an extensive simulation study, the performance of our approach on a number of benchmark functions is shown to be at least equivalent, but mostly better than previous approaches and fits both functions of smoothly varying complexity and discontinuous functions well. Results from two applications also reflecting these two situations support the simulation results
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
Meta-analysis of functional neuroimaging data using Bayesian nonparametric binary regression
In this work we perform a meta-analysis of neuroimaging data, consisting of
locations of peak activations identified in 162 separate studies on emotion.
Neuroimaging meta-analyses are typically performed using kernel-based methods.
However, these methods require the width of the kernel to be set a priori and
to be constant across the brain. To address these issues, we propose a fully
Bayesian nonparametric binary regression method to perform neuroimaging
meta-analyses. In our method, each location (or voxel) has a probability of
being a peak activation, and the corresponding probability function is based on
a spatially adaptive Gaussian Markov random field (GMRF). We also include
parameters in the model to robustify the procedure against miscoding of the
voxel response. Posterior inference is implemented using efficient MCMC
algorithms extended from those introduced in Holmes and Held [Bayesian Anal. 1
(2006) 145--168]. Our method allows the probability function to be locally
adaptive with respect to the covariates, that is, to be smooth in one region of
the covariate space and wiggly or even discontinuous in another. Posterior
miscoding probabilities for each of the identified voxels can also be obtained,
identifying voxels that may have been falsely classified as being activated.
Simulation studies and application to the emotion neuroimaging data indicate
that our method is superior to standard kernel-based methods.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS523 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Function estimation with locally adaptive dynamic models
We present a nonparametric Bayesian method for fitting unsmooth and highly oscillating functions, which is based on a locally adaptive hierarchical extension of standard dynamic or state space models. The main idea is to introduce locally varying variances in the state equations and to add a further smoothness prior for this variance function. Estimation is fully Bayesian and carried out by recent MCMC techniques. The whole approach can be understood as an alternative to other nonparametric function estimators, such as local or penalized regression with variable bandwidth or smoothing parameter selection. Performance is illustrated with simulated data, including unsmooth examples constructed for wavelet shrinkage, and by an application to sales data. Although the approach is developed for classical Gaussian nonparametric regression, it can be extended to more complex regression problems
Locally adaptive smoothing with Markov random fields and shrinkage priors
We present a locally adaptive nonparametric curve fitting method that
operates within a fully Bayesian framework. This method uses shrinkage priors
to induce sparsity in order-k differences in the latent trend function,
providing a combination of local adaptation and global control. Using a scale
mixture of normals representation of shrinkage priors, we make explicit
connections between our method and kth order Gaussian Markov random field
smoothing. We call the resulting processes shrinkage prior Markov random fields
(SPMRFs). We use Hamiltonian Monte Carlo to approximate the posterior
distribution of model parameters because this method provides superior
performance in the presence of the high dimensionality and strong parameter
correlations exhibited by our models. We compare the performance of three prior
formulations using simulated data and find the horseshoe prior provides the
best compromise between bias and precision. We apply SPMRF models to two
benchmark data examples frequently used to test nonparametric methods. We find
that this method is flexible enough to accommodate a variety of data generating
models and offers the adaptive properties and computational tractability to
make it a useful addition to the Bayesian nonparametric toolbox.Comment: 38 pages, to appear in Bayesian Analysi
A Semi-parametric Technique for the Quantitative Analysis of Dynamic Contrast-enhanced MR Images Based on Bayesian P-splines
Dynamic Contrast-enhanced Magnetic Resonance Imaging (DCE-MRI) is an
important tool for detecting subtle kinetic changes in cancerous tissue.
Quantitative analysis of DCE-MRI typically involves the convolution of an
arterial input function (AIF) with a nonlinear pharmacokinetic model of the
contrast agent concentration. Parameters of the kinetic model are biologically
meaningful, but the optimization of the non-linear model has significant
computational issues. In practice, convergence of the optimization algorithm is
not guaranteed and the accuracy of the model fitting may be compromised. To
overcome this problems, this paper proposes a semi-parametric penalized spline
smoothing approach, with which the AIF is convolved with a set of B-splines to
produce a design matrix using locally adaptive smoothing parameters based on
Bayesian penalized spline models (P-splines). It has been shown that kinetic
parameter estimation can be obtained from the resulting deconvolved response
function, which also includes the onset of contrast enhancement. Detailed
validation of the method, both with simulated and in vivo data, is provided
Stochastic expansions using continuous dictionaries: L\'{e}vy adaptive regression kernels
This article describes a new class of prior distributions for nonparametric
function estimation. The unknown function is modeled as a limit of weighted
sums of kernels or generator functions indexed by continuous parameters that
control local and global features such as their translation, dilation,
modulation and shape. L\'{e}vy random fields and their stochastic integrals are
employed to induce prior distributions for the unknown functions or,
equivalently, for the number of kernels and for the parameters governing their
features. Scaling, shape, and other features of the generating functions are
location-specific to allow quite different function properties in different
parts of the space, as with wavelet bases and other methods employing
overcomplete dictionaries. We provide conditions under which the stochastic
expansions converge in specified Besov or Sobolev norms. Under a Gaussian error
model, this may be viewed as a sparse regression problem, with regularization
induced via the L\'{e}vy random field prior distribution. Posterior inference
for the unknown functions is based on a reversible jump Markov chain Monte
Carlo algorithm. We compare the L\'{e}vy Adaptive Regression Kernel (LARK)
method to wavelet-based methods using some of the standard test functions, and
illustrate its flexibility and adaptability in nonstationary applications.Comment: Published in at http://dx.doi.org/10.1214/11-AOS889 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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