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Bayesian Methods. Will Penny and Guillaume Flandin. Wellcome Department of Imaging Neuroscience, University College London, UK. SPM Course, London, May 12 th 2006. Overview. Bayes rule and model comparison ANOVAs Normalisation Segmentation fMRI stats Hemodynamic models
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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 Y Eliminating p(Y,) gives Bayes rule Likelihood Prior Posterior Evidence
Gaussian Likelihood and Prior Posterior Posterior Likelihood Prior Relative Precision Weighting
Select the model m with the highest probability given the data: Model comparison and Bayes factor: Model Comparison Model evidence (marginal likelihood): Accuracy Complexity
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)
Squared distance between parameters and their expected values (Prior) Mean square difference between template and source image (Likelihood) Spatial Normalisation Posterior Deformation parameters Template 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+Ewhere E is an AR(p) a l b A Y
Spatial prior Over the regression coefficients: Spatial precison: determines the amount of smoothness Shrinkage prior Spatial kernel matrix Gaussian Markov Random Field priors 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 Convergence F Sensitivity o Global o Spatialo Smoothing Iteration Number 1-Specificity
Global prior Spatial Prior Event related fMRI: familiar versus unfamiliar faces Smoothing
Posterior Probability Maps Posterior distribution: probability of getting an effect, given the data mean: size of effectprecision: variability Posterior probability map: images of the probability or confidence that an activation exceeds some specified threshold, given the data • Two thresholds: • activation threshold : percentage of whole brain mean signal (physiologically relevant size of effect) • probability that voxels must exceed to be displayed (e.g. 95%)
Posterior Probability Maps Activation threshold Mean (Cbeta_*.img) PPM (spmP_*.img) Posterior probability distribution p(b |Y) Probability Std dev (SDbeta_*.img)
Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)
Hemodynamic basis sets Informed Gamma Fourier
FIR models Size of signal Time after event 5s
Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)
SPC V1 V5 Dynamic Causal Models m=2 m=1 Photic Photic SPC 0.85 0.86 0.75 0.70 0.84 1.42 0.89 Attention V1 0.55 -0.02 -0.02 0.57 0.56 V5 Motion Motion 0.58 Attention m=3 Photic SPC Bayesian Evidence: 0.85 0.70 0.85 1.36 Attention V1 0.03 Bayes factors: -0.02 0.57 V5 Motion 0.23 Attention
Summary • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)