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

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

Bayes Rule

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

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

General Case: Arbitrary Error Covariances EM algorithm E-Step ( ) y C X T 1 - = e q h M-Step r for i and j { } { Q tr J g i j ij k å + l Friston, K. et al. (2002), Neuroimage

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 2nd level ie. RFX

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

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

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

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

Bayesian Inference: Posterior Probability Maps PPMs Posterior Likelihood Prior SPMs

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

PPMs Advantages Disadvantages Use of Shrinkage One can infer a cause priors over voxels is computationally demanding Utility of Bayesian approach is yet to be established 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

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

Extension to a MIMO system The bilinear model neuronal changes intrinsic connectivity induced response Input u(t) activity x1(t) x3(t) x2(t) hemodynamics response y(t)=(X) Hemodynamic model

Dynamical Causal Models Functional integration and the modulation of specific pathways V1 V4 BA37 STG BA39 Cognitive set - u2(t) {e.g. semantic processing} Stimuli - u1(t) {e.g. visual words}