Spatial Bayesian Variable Selection With Application to Functional Magnetic Resonance Imaging

We propose a procedure to undertake Bayesian variable selection and model averaging for a series of regressions located on a lattice. For those regressors that are in common in the regressions, we consider using an Ising prior to smooth spatially the indicator variables representing whether or not the variable is zero or nonzero in each regression. This smooths spatially the probabilities that each independent variable is nonzero in each regression and indirectly smooths spatially the regression coefficients. We discuss how single-site sampling schemes can be used to evaluate the joint posterior distribution. The approach is applied to the problem of functional magnetic resonance imaging in medical statistics, where massive datasets arise that require prompt processing. Here the Ising prior with a three-dimensional neighborhood structure is used to smooth spatially activation maps from regression models of blood oxygenation. The Ising prior also has the advantage of allowing incorporation of anatomic prior information through the external field. Using a visual experiment, we show how a single-site sampling scheme can provide rapid evaluation of the posterior activation maps and activation amplitudes. The approach is shown to result in maps that are superior to those produced by a recent Bayesian approach using a continuous Markov random field for the activation amplitude.

[1]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[2]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[3]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[4]  T. J. Mitchell,et al.  Bayesian Variable Selection in Linear Regression , 1988 .

[5]  Andrew L. Rukhin,et al.  Tools for statistical inference , 1991 .

[6]  G. Parisi,et al.  Simulated tempering: a new Monte Carlo scheme , 1992, hep-lat/9205018.

[7]  E. George,et al.  Journal of the American Statistical Association is currently published by American Statistical Association. , 2007 .

[8]  C. Geyer.  Estimating Normalizing Constants and Reweighting Mixtures , 1994 .

[9]  Walter R. Gilks,et al.  Hypothesis testing and model selection , 1995 .

[10]  Karl J. Friston,et al.  Spatial registration and normalization of images , 1995 .

[11]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[12]  Jonathan D. Cohen,et al.  Improved Assessment of Significant Activation in Functional Magnetic Resonance Imaging (fMRI): Use of a Cluster‐Size Threshold , 1995, Magnetic resonance in medicine.

[13]  M. Clyde,et al.  Prediction via Orthogonalized Model Mixing , 1996 .

[14]  R. Kohn,et al.  Nonparametric regression using Bayesian variable selection , 1996 .

[15]  Adrian E. Raftery,et al.  Hypothesis testing and model selection , 1996 .

[16]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[17]  D. Madigan,et al.  Bayesian Model Averaging for Linear Regression Models , 1997 .

[18]  R. Kohn,et al.  A Bayesian approach to nonparametric bivariate regression , 1997 .

[19]  Adrian F. M. Smith,et al.  Automatic Bayesian curve fitting , 1998 .

[20]  D. Higdon.  Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications , 1998 .

[21]  Xiao-Li Meng,et al.  Simulating Normalizing Constants: From Importance Sampling to Bridge Sampling to Path Sampling , 1998 .

[22]  D. Denison,et al.  Bayesian model order determination and basis selectionfor seemingly unrelated regressionsC , 1999 .

[23]  Christopher R. Genovese,et al.  A Bayesian Time-Course Model for Functional Magnetic Resonance Imaging Data , 2000 .

[24]  C Gössl,et al.  Dynamic models in fMRI , 2000, Magnetic resonance in medicine.

[25]  R. Kohn,et al.  Nonparametric seemingly unrelated regression , 2000 .

[26]  L. Fahrmeir,et al.  Bayesian spatio-temporal inference in functional magnetic resonance imaging , 2001 .

[27]  Christoff Gössl.  Bayesian Models in Functional Magnetic Resonance Imaging: Approaches for Human Brain Mapping , 2001 .

[28]  Robert Kohn,et al.  Nonparametric regression using linear combinations of basis functions , 2001, Stat. Comput..

[29]  R. Kohn,et al.  Statistical Correction of a Deterministic Numerical Weather Prediction Model , 2001 .

[30]  Ludwig Fahrmeir,et al.  Bayesian varying-coefficient models using adaptive regression splines , 2001 .

[31]  Ludwig Fahrmeir,et al.  Semiparametric Bayesian models for human brain mapping , 2002 .

[32]  Bani K. Mallick,et al.  Accounting for Model Uncertainty in Seemingly Unrelated Regressions , 2002 .

[33]  T. Fearn,et al.  Bayes model averaging with selection of regressors , 2002 .

[34]  P. Green,et al.  Hidden Markov Models and Disease Mapping , 2002 .

[35]  Ludwig Fahrmeir,et al.  Assessing brain activity through spatial bayesian variable selection , 2003, NeuroImage.

[36]  M. Clyde,et al.  Model Uncertainty , 2003 .

[37]  Mark W. Woolrich,et al.  Fully Bayesian spatio-temporal modeling of FMRI data , 2004, IEEE Transactions on Medical Imaging.

[38]  P. Green,et al.  Bayesian Variable Selection and the Swendsen-Wang Algorithm , 2004 .

[39]  Adrian E. Raftery,et al.  Bayesian model averaging: development of an improved multi-class, gene selection and classification tool for microarray data , 2005, Bioinform..

[40]  Karl J. Friston,et al.  Bayesian fMRI time series analysis with spatial priors , 2005, NeuroImage.

[41]  M. Smith,et al.  Estimation of Binary Markov Random Fields Using Markov chain Monte Carlo , 2006 .

[42]  L. Fahrmeir,et al.  Adaptive Gaussian Markov random fields with applications in human brain mapping , 2007 .