Bayesian Inference and Posterior Probability Maps

1. Bayesian Inference and Posterior Probability Maps Guillaume Flandin Wellcome Department of Imaging Neuroscience, University College London, UK SPM Course, London, May 2005

2. Overview • Introduction • Bayesian inference • Segmentation and Normalisation • Gaussian Prior and Likelihood • Posterior Probability Maps (PPMs) • Global shrinkage prior (2nd level) • Spatial prior (1st level) • Bayesian Model Comparison • Comparing Dynamic Causal Models (DCMs) • Summary SPM5

3. Classical approach shortcomings In SPM, the p-value reflects the probability of getting the observed data in the effect’s absence. If sufficiently small, this p-value can be used to reject the null hypothesis that the effect is negligible. Probability of the data, given no activation • One can never accept the null hypothesis Shortcomings of this approach: • Given enough data, one can always demonstrate a significant effect at every voxel • Correction for multiple comparisons Solution: using the probability distribution of the activation given the data.

4. Baye’s Rule Given p(Y), p() and p(Y,) Conditional densities are given by  Y Eliminating p(Y,) gives Baye’s rule Likelihood Prior Posterior Evidence

5. Formulation of a generative model  likelihood p(Y|) prior distribution p() • Observation of data Y • Update of beliefs based upon observations, given a prior state of knowledge Bayesian Inference Three steps:

6. Overview • Introduction • Bayesian inference • Segmentation and Normalisation • Gaussian Prior and Likelihood • Posterior Probability Maps (PPMs) • Global shrinkage prior (2nd level) • Spatial prior (1st level) • Bayesian Model Comparison • Comparing Dynamic Causal Models (DCMs) • Summary SPM5

7. Bayes and Spatial Preprocessing Normalisation Deformation parameters MAP: Squared distance between parameters and their expected values (regularisation) Mean square difference between template and source image (goodness of fit) Unlikely deformation

8. Bayes and Spatial Preprocessing 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.

9. Overview • Introduction • Bayesian inference • Segmentation and Normalisation • Gaussian Prior and Likelihood • Posterior Probability Maps (PPMs) • Global shrinkage prior (2nd level) • Spatial prior (1st level) • Bayesian Model Comparison • Comparing Dynamic Causal Models (DCMs) • Summary SPM5

10. Gaussian Case Likelihood and Prior Posterior Posterior Likelihood Prior Relative Precision Weighting

11. Overview • Introduction • Bayesian inference • Segmentation and Normalisation • Gaussian Prior and Likelihood • Posterior Probability Maps (PPMs) • Global shrinkage prior (2nd level) • Spatial prior (1st level) • Bayesian Model Comparison • Comparing Dynamic Causal Models (DCMs) • Summary SPM5

12. Bayesian fMRI General Linear Model: with What are the priors? • In “classical” SPM, no (flat) priors • In “full” Bayes, priors might be from theoretical arguments or from independent data • In “empirical” Bayes, priors derive from the same data, assuming a hierarchical model for generation of the data Parameters of one level can be made priors on distribution of parameters at lower level Parameters and hyperparameters at each level can be estimated using EM algorithm

13. Bayesian fMRI: what prior? General Linear Model: Shrinkage prior: If Cε and Cβ are known, then the posterior is: 0 In the absence of evidence to the contrary, parameters will shrink to zero • We are looking for the same effect over multiple voxels • Pooled estimation of Cβover voxels via EM

14. Bayesian Inference PPMs Posterior Likelihood Prior SPMs Bayesian test Classical T-test

15. 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%)

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

17. SPM and PPM (PET) PPMs: Show activations greater than a given size SPMs: Show voxels with non-zeros activations Verbal fluency data

18. SPM and PPM (fMRI 1st level) PPM SPM C. Buchel et al, Cerebral Cortex, 1997

19. SPM and PPM (fMRI 2nd level) PPM SPM R. Henson et al, Cerebral Cortex, 2002

20. PPMs: Pros and Cons Disadvantages Advantages • One can infer a cause DID NOT elicit a response • Inference independent of search volume • SPMs conflate effect-size and effect-variability • Use of priors over voxels is computationally demanding • Utility of Bayesian approach is yet to be established

21. Bayesian fMRI with spatial priors 1st level 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  Automatically spatially regularisation of Regression coefficients and AR coefficients

22. The 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

23. 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.

24. Prior, Likelihood and Posterior The prior: The likelihood: The posterior? p(b |Y) ? The posterior over b doesn’t factorise over k or n.  Exact inference is intractable.

25. Variational Bayes Approximate posteriors that allows for factorisation Variational Bayes Algorithm Initialisation While (ΔF > tol) Update Suff. Stats. for β Update Suff. Stats. for A Update Suff. Stats. for λ Update Suff. Stats. for α Update Suff. Stats. for γ End

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

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

28. SPM5 Interface

29. Overview • Introduction • Bayesian inference • Segmentation and Normalisation • Gaussian Prior and Likelihood • Posterior Probability Maps (PPMs) • Global shrinkage prior (2nd level) • Spatial prior (1st level) • Bayesian Model Comparison • Comparing Dynamic Causal Models (DCMs) • Summary SPM5

30. Select the model m with the highest probability given the data: Model comparison and Baye’s factor: Bayesian Model Comparison Model evidence (marginal likelihood): Accuracy Complexity

31. SPC V1 V5 Comparing 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

32. Summary • Bayesian inference: • There is no null hypothesis • Allows to incorporate some prior belief • Posterior Probability Maps • Empirical Bayes for 2nd level analyses: • Global shrinkage prior • Variational Bayes for single-subject analyses: • Spatial prior on regression and AR coefficients • Bayesian framework also allows: • Bayesian Model Comparison

33. References • Classical and Bayesian Inference, Penny and Friston, Human Brain Function (2nd edition), 2003. • Classical and Bayesian Inference in Neuroimaging: Theory/Applications, Friston et al., NeuroImage, 2002. • Posterior Probability Maps and SPMs, Friston and Penny, NeuroImage, 2003. • Variational Bayesian Inference for fMRI time series, Penny et al., NeuroImage, 2003. • Bayesian fMRI time series analysis with spatial priors, Penny et al., NeuroImage, 2005. • Comparing Dynamic Causal Models, Penny et al, NeuroImage, 2004.