Bayesian Methods Will Penny and Guillaume Flandin Wellcome Department of Imaging Neuroscience, University College London, UK SPM Course, London, May 12th 2006 Overview Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models Connectivity models (DCM) Overview Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models Connectivity models (DCM) Bayes rule Given p(Y), p() and p(Y,) Conditional densities are given by p (Y , ) p ( | Y ) Y p (Y | ) p (Y ) p (Y , ) p ( ) Eliminating p(Y,) gives Bayes rule Likelihood Posterior p ( | Y ) Prior p (Y | ) p ( ) p (Y ) Evidence Gaussian Likelihood and Prior N , N , p p y | (1 ) 1 (1 ) (1 ) (1 ) 1 (2) (2) Posterior Posterior p (1 ) Likelihood | y N m, p 1 Prior p (1 ) ( 2 ) m (1 ) p (1 ) (2) (2) p Relative Precision Weighting (2) m (1 ) Bivariate Gaussian Model Comparison Select the model m with the highest probability given the data: p (m | Y ) P (Y | m ) p ( m ) p (Y ) Model evidence (marginal likelihood): p (Y | m ) p (Y | m , m Accuracy ) p ( m | m ) d m Complexity Model comparison and Bayes factor: B12 p (Y | m 1 ) p (Y | m 2 ) B12 p(m1|Y) Evidence 1 to 3 50-75 Weak 3 to 20 75-95 Positive 20 to 150 95-99 Strong 150 99 Very strong Overview Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models Connectivity models (DCM) ANOVA Four conditions Model 0 (dotted lines) – no effect Model 1 (solid lines) – an effect Compare inferences on 100 data sets p=0.05 Bayesian BF=3 Classical Overview Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models Connectivity models (DCM) Spatial Normalisation Posterior Deformation parameters log p ( | y ) log p ( y | ) log p ( ) Mean square difference between template and source image (Likelihood) Template Squared distance between parameters and their expected values (Prior) Max Likelihood Max Posterior Segmentation Intensities are modelled by a mixture of K Gaussian distributions. Overlay prior belonging probability maps to assist the segmentation: Prior probability of each voxel being of a particular type is derived from segmented images of 151 subjects. Overview Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models Connectivity models (DCM) fMRI stats Even without applied spatial smoothing, activation maps (and maps of eg. AR coefficients) have spatial structure. Contrast AR(1) Definition of a spatial prior via Gaussian Markov Random Field Automatic spatial regularisation of Regression coefficients and AR coefficients Generative Model General Linear Model with Auto-Regressive error terms (GLM-AR): Y=X β +E where E is an AR(p) a 1 p ( b k ) N (0, a k D b 1 p (a p ) N (0, ) 1 p D A p Y yt X t b ae i i 1 t i t 1 ) Spatial prior Over the regression coefficients: p b N 0 , a k Shrinkage prior 1 k D 1 Spatial precison: determines the amount of smoothness Spatial kernel matrix Gaussian Markov Random Field priors D 1 D 1 d ji d ij 1 1 1 on diagonal elements dii dij > 0 if voxels i and j are neighbors. 0 elsewhere Same prior on the AR coefficients. Convergence & Sensitivity ROC curve Sensitivity Convergence F Iteration Number o Global o Spatial o Smoothing 1-Specificity Event related fMRI: familiar versus unfamiliar faces Smoothing Global prior Spatial Prior Posterior Probability Maps Posterior distribution: probability of getting an effect, given the data p(b | y) mean: size of effect precision: variability Posterior probability map: images of the probability or confidence that an activation exceeds some specified threshold, given the data p(b | y) a p(b | y) b Two thresholds: • activation threshold : percentage of whole brain mean signal (physiologically relevant size of effect) • probability a that voxels must exceed to be displayed (e.g. 95%) Posterior Probability Maps Activation threshold p(b | y) a Mean (Cbeta_*.img) Posterior probability distribution p(b |Y) Probability a Std dev (SDbeta_*.img) PPM (spmP_*.img) SPM5 Interface Overview Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models Connectivity models (DCM) Fourier Gamma Informed Hemodynamic basis sets FIR models Size of signal 5s Time after event Inf2: Canonical + temporal deriv Inf2: Canonical + temporal deriv SPM5: from spm_vb_roi_basis.m Overview Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models Connectivity models (DCM) Dynamic Causal Models m=2 m=1 Photic SPC 0.85 Photic SPC 0.86 0.70 V1 V1 0.57 0.75 0.84 0.89 Attention 0.55 -0.02 Motion 0.58 Attention 1.42 V5 -0.02 0.56 V5 Motion m=3 Bayesian Evidence: Photic SPC 0.85 0.70 0.85 1.36 V1 Bayes factors: 0.03 0.57 -0.02 0.23 Motion Attention V5 Attention Summary Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models Connectivity models (DCM)

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