Bayesian Inference in SPM2 Will Penny K. Friston, J. Ashburner, J.-B. Poline, R. Henson, S. Kiebel, D. Glaser Wellcome Department of Imaging Neuroscience, University College London, UK

SPM99 RealignmentSmoothing Normalisation General linear model Statistical parametric map (SPM) fMRI time-series Parameter estimates Design matrix Template Kernel p <0.05 Inference with Gaussian field theory Adjusted regional data spatial modes and effective connectivity

What’s new in SPM2 ? §Spatial transformation of images §Batch Mode §Modelling and Inference Expectation-Maximisation (EM) Restricted Maximum Likelihood (ReML) Parametric Empirical Bayes (PEB)

Hierarchical model Single-level model Parametric Empirical Bayes (PEB) Restricted Maximimum Likelihood (ReML) Hierarchical models

Bayes Rule

Example 2:Univariate model Likelihood and Prior Posterior Relative Precision Weighting

Example 2:Multivariate two-level model Likelihood and Prior Assume diagonal precisions PosteriorPrecisions Data-determined parameters Assume Shrinkage Prior

General Case: Arbitrary Error Covariances E-Step  yCXC XCXC T yy T y           M-Step y Xyr   for i and j { }{ }{}{         CQCQtrJ XCQCXC rCQCrCQ g ijij i T y i T ii } kk QCC gJ      1 Friston, K. et al. (2002), Neuroimage EM algorithm

Pooling assumption Decompose error covariance at each voxel, i, into a voxel specific term, r(i), and voxel-wide terms.

What’s new in SPM2 ? §Corrections for Non-Sphericity §Posterior Probability Maps (PPMs) §Haemodynamic modelling §Dynamic Causal Modelling (DCM)

Non-sphericity §Relax assumption that errors are Independent and Identically Distributed (IID) §Non-independent errors eg. repeated measures within subject §Non-identical errors eg. unequal condition/subject error variances §Correlation in fMRI time series §Allows multiple parameters at 2 nd level ie. RFX

Single-subject contrasts from Group FFX PET Verbal Fluency SPMs,p<0.001 uncorrected SphericityNon-sphericity Non-identical error variances

Correlation in fMRI time series Model errors for each subject as AR(1) + white noise.

The Interface OLS Parameters, REML Hyperparameters PEB Parameters and Hyperparameters No Priors Shrinkage priors

Bayesian estimation: Two-level model 1 st level = within-voxel 2nd level = between-voxels Likelihood Shrinkage Prior In the absence of evidence to the contrary parameters will shrink to zero

LikelihoodPrior Posterior SPMs PPMs Bayesian Inference: Posterior Probability Maps

SPMs and PPMs PPMs: Show activations of a given size SPMs: show voxels with non-zero activations

PPMs AdvantagesDisadvantages One can infer a cause DID NOT elicit a response SPMs conflate effect-size and effect-variability No multiple comparisons problem (hence no smoothing) P-values don’t change with search volume Use of Shrinkage priors over voxels is computationally demanding Utility of Bayesian approach is yet to be established

The Interface Hemodynamic Modelling

The hemodynamic model

Hemodynamics

Inference with MISO models FUNCTIONAL SEGREGATION: This voxel IS NOT responsive to attention

The Interface Dynamic Causal Modelling

hemodynamics response y(t)= (X) response y(t)= (X) hemodynamics response y(t)= (X) hemodynamics Hemodynamic model Extension to a MIMO system Input u(t) activity x 1 (t) activity x 3 (t) activity x 2 (t) The bilinear model neuronal changes intrinsic connectivity induced response induced connectivity

V1 V4 BA37 STG BA39 Cognitive set - u 2 (t) {e.g. semantic processing} Stimuli - u 1 (t) {e.g. visual words} Dynamical Causal Models Functional integration and the modulation of specific pathways

Summary §SPM2 – modelling and inference §Hierarchical Models and EM §Corrections for Non-Sphericity §Posterior Probability Maps (PPMs) §Haemodynamic modelling §Dynamic Causal Modelling (DCM)