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Bayesian Methods

This comprehensive overview covers Bayesian methods, model comparison, ANOVAs, normalization, segmentation, fMRI statistics, hemodynamic models, and connectivity models (DCM) in neuroimaging. Topics include Bayes rule, Gaussian likelihood and prior, model evidence, spatial normalization, posterior probability maps, hemodynamic basis sets, and dynamic causal models (DCM).

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Bayesian Methods

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  1. Bayesian Methods Will Penny and Guillaume Flandin Wellcome Department of Imaging Neuroscience, University College London, UK SPM Course, London, May 12th 2006

  2. Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)

  3. Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)

  4. 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

  5. Gaussian Likelihood and Prior Posterior Posterior Likelihood Prior Relative Precision Weighting

  6. Bivariate Gaussian

  7. Select the model m with the highest probability given the data: Model comparison and Bayes factor: Model Comparison Model evidence (marginal likelihood): Accuracy Complexity

  8. Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)

  9. ANOVA Four conditions Model 0 (dotted lines) – no effect Model 1 (solid lines) – an effect

  10. Compare inferences on 100 data sets p=0.05 Bayesian BF=3 Classical

  11. Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)

  12. 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

  13. 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.

  14. Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)

  15. 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

  16. 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

  17. 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.

  18. Convergence & Sensitivity ROC curve Convergence F Sensitivity o Global o Spatialo Smoothing Iteration Number 1-Specificity

  19. Global prior Spatial Prior Event related fMRI: familiar versus unfamiliar faces Smoothing

  20. 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%)

  21. Posterior Probability Maps Activation threshold  Mean (Cbeta_*.img) PPM (spmP_*.img) Posterior probability distribution p(b |Y) Probability  Std dev (SDbeta_*.img)

  22. SPM5 Interface

  23. Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)

  24. Hemodynamic basis sets Informed Gamma Fourier

  25. FIR models Size of signal Time after event 5s

  26. Inf2: Canonical + temporal deriv

  27. Inf2: Canonical + temporal deriv

  28. SPM5: from spm_vb_roi_basis.m

  29. Overview • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)

  30. 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

  31. Summary • Bayes rule and model comparison • ANOVAs • Normalisation • Segmentation • fMRI stats • Hemodynamic models • Connectivity models (DCM)

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