1. Bayesian models for fMRI data Methods & models for fMRI data analysis 06 May 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London With many thanks for slides & images to: FIL Methods group, particularly Guillaume Flandin The Reverend Thomas Bayes (1702-1761)

2. Why do I need to learn about Bayesian stats? Because SPM is getting more and more Bayesian: Segmentation & spatial normalisation Posterior probability maps (PPMs) –1 st level: specific spatial priors –2 nd level: global spatial priors Dynamic Causal Modelling (DCM) Bayesian Model Selection (BMS) EEG: source reconstruction

3. RealignmentSmoothing Normalisation General linear model Statistical parametric map (SPM) Image time-series Parameter estimates Design matrix Template Kernel Gaussian field theory p <0.05 Statisticalinference Bayesian segmentation and normalisation Bayesian segmentation and normalisation Spatial priors on activation extent Spatial priors on activation extent Posterior probability maps (PPMs) Posterior probability maps (PPMs) Dynamic Causal Modelling Dynamic Causal Modelling

4. p-value: probability of getting the observed data in the effect’s absence. If small, reject null hypothesis that there is no effect. Limitations:  One can never accept the null hypothesis  Given enough data, one can always demonstrate a significant effect  Correction for multiple comparisons necessary Solution: infer posterior probability of the effect Probability of observing the data y, given no effect (  = 0). Problems of classical (frequentist) statistics Probability of the effect, given the observed data

5. Overview of topics Bayes' rule Bayesian update rules for Gaussian densities Bayesian analyses in SPM –Segmentation & spatial normalisation –Posterior probability maps (PPMs) 1st level: specific spatial priors 2nd level: global spatial priors –Bayesian Model Selection (BMS)

6. Bayesian statistics posterior  likelihood ∙ prior Bayes theorem allows one to formally incorporate prior knowledge into computing statistical probabilities. Priors can be of different sorts: empirical, principled or shrinkage priors. The “posterior” probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision. new data prior knowledge

7. Bayes in motion - an animation

8. Given data y and parameters , the conditional probabilities are: Eliminating p(y,  ) gives Bayes’ rule: Likelihood Prior Evidence Posterior Bayes’ rule

9. y y  Observation of data likelihood p(y|  ) prior distribution p(  ) likelihood p(y|  ) prior distribution p(  )  Formulation of a generative model  Update of beliefs based upon observations, given a prior state of knowledge Principles of Bayesian inference

10. Likelihood & Prior Posterior: Posterior mean = variance-weighted combination of prior mean and data mean Prior Likelihood Posterior Posterior mean & variance of univariate Gaussians

11. Likelihood & prior Posterior: Prior Likelihood Posterior Same thing – but expressed as precision weighting Relative precision weighting

12. Likelihood & Prior Posterior Relative precision weighting Prior Likelihood Posterior Same thing – but explicit hierarchical perspective

13. Bayesian GLM: univariate case Relative precision weighting Normal densities Univariate linear model

14. One step if C e is known. Otherwise iterative estimation with EM. General Linear Model Bayesian GLM: multivariate case Normal densities 22 11

15. An intuitive example

16. Less intuitive

17. Even less intuitive

18. Likelihood distributions from different subjects are independent  one can use the posterior from one subject as the prior for the next Under Gaussian assumptions this is easy to compute: group posterior covariance individual posterior covariances group posterior mean individual posterior covariances and means “Today’s posterior is tomorrow’s prior” Bayesian (fixed effects) group analysis

19. Bayesian analyses in SPM5 Segmentation & spatial normalisation Posterior probability maps (PPMs) –1 st level: specific spatial priors –2 nd level: global spatial priors Dynamic Causal Modelling (DCM) Bayesian Model Selection (BMS) EEG: source reconstruction

20. Spatial normalisation: Bayesian regularisation Deformations consist of a linear combination of smooth basis functions  lowest frequencies of a 3D discrete cosine transform. Find maximum a posteriori (MAP) estimates: simultaneously minimise –squared difference between template and source image –squared difference between parameters and their priors MAP: Deformation parameters “Difference” between template and source image Squared distance between parameters and their expected values (regularisation)

21. Bayesian segmentation with empirical priors Goal: for each voxel, compute probability that it belongs to a particular tissue type, given its intensity Likelihood model: Intensities are modelled by a mixture of Gaussian distributions representing different tissue classes (e.g. GM, WM, CSF). Priors are obtained from tissue probability maps (segmented images of 151 subjects). Goal: for each voxel, compute probability that it belongs to a particular tissue type, given its intensity Likelihood model: Intensities are modelled by a mixture of Gaussian distributions representing different tissue classes (e.g. GM, WM, CSF). Priors are obtained from tissue probability maps (segmented images of 151 subjects). Ashburner & Friston 2005, NeuroImage p (tissue | intensity)  p (intensity | tissue) ∙ p (tissue)

22. Unified segmentation & normalisation Circular relationship between segmentation & normalisation: –Knowing which tissue type a voxel belongs to helps normalisation. –Knowing where a voxel is (in standard space) helps segmentation. Build a joint generative model: –model how voxel intensities result from mixture of tissue type distributions –model how tissue types of one brain have to be spatially deformed to match those of another brain Using a priori knowledge about the parameters: adopt Bayesian approach and maximise the posterior probability Ashburner & Friston 2005, NeuroImage

23. General Linear Model: What are the priors? with In “classical” SPM, no priors (= “flat” priors) Full Bayes: priors are predefined on a principled or empirical basis Empirical Bayes: priors are estimated from the data, assuming a hierarchical generative model  PPMs in SPM Parameters of one level = priors for distribution of parameters at lower level Parameters and hyperparameters at each level can be estimated using EM Bayesian fMRI analyses

24. Hierarchical models and Empirical Bayes Hierarchical model Hierarchical model Parametric Empirical Bayes (PEB) Parametric Empirical Bayes (PEB) EM = PEB = ReML Restricted Maximum Likelihood (ReML) Restricted Maximum Likelihood (ReML) Single-level model Single-level model

25. Posterior Probability Maps (PPMs) Posterior distribution: probability of the effect given the data Posterior probability map: images of the probability (confidence) that an activation exceeds some specified threshold , given the data y 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%) 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%) mean: size of effect precision: variability

26. PPMs vs. SPMs LikelihoodPrior Posterior SPMsSPMs PPMsPPMs Bayesian test: Classical t-test:

27. 2 nd level PPMs with global priors In the absence of evidence to the contrary, parameters will shrink to zero. In the absence of evidence to the contrary, parameters will shrink to zero. 1 st level (GLM): 2 nd level (shrinkage prior): 0 Basic idea: use the variance of  over voxels as prior variance of  at any particular voxel. 2 nd level:  (2) = average effect over voxels,  (2) = voxel-to-voxel variation.  (1) reflects regionally specific effects  assume that it sums to zero over all voxels  shrinkage prior at the second level  variance of this prior is implicitly estimated by estimating  (2)

28. Shrinkage Priors Small & variable effect Large & variable effect Small but clear effect Large & clear effect

29. 2 nd level PPMs with global priors 1 st level (GLM): 2 nd level (shrinkage prior): Once C ε and C  are known, we can apply the usual rule for computing the posterior mean & covariance:  We are looking for the same effect over multiple voxels  Pooled estimation of C  over voxels voxel-specific global  pooled estimate Friston & Penny 2003, NeuroImage

30. PPMs and multiple comparisons No need to correct for multiple comparisons: Thresholding a PPM at 95% confidence: in every voxel, the posterior probability of an activation   is  95%. At most, 5% of the voxels identified could have activations less than . Independent of the search volume, thresholding a PPM thus puts an upper bound on the false discovery rate.

31. PPMs vs.SPMs PPMs: Show activations greater than a given size SPMs: Show voxels with non-zero activations

32. PPMs: pros and cons One can infer that a cause did not elicit a response Inference is independent of search volume SPMs conflate effect- size and effect- variability One can infer that a cause did not elicit a response Inference is independent of search volume SPMs conflate effect- size and effect- variability Disadvantages Advantages Estimating priors over voxels is computationally demanding Practical benefits are yet to be established Thresholds other than zero require justification Estimating priors over voxels is computationally demanding Practical benefits are yet to be established Thresholds other than zero require justification

33. 1 st level PPMs with local spatial priors Neighbouring voxels often not independent Spatial dependencies vary across the brain But spatial smoothing in SPM is uniform Matched filter theorem: SNR maximal when smoothing the data with a kernel which matches the smoothness of the true signal Basic idea: estimate regional spatial dependencies from the data and use this as a prior in a PPM  regionally specific smoothing  markedly increased sensitivity Contrast map AR(1) map Penny et al. 2005, NeuroImage

34.  A q1q1 q2q2  W Y u1u1 u2u2 Y=XW+E r1r1 r2r2 The generative spatio-temporal model Penny et al. 2005, NeuroImage  = spatial precision of parameters = observation noise precision  = precision of AR coefficients

35. Prior for k-th parameter: Shrinkage prior Spatial kernel matrix Spatial precision: determines the amount of smoothness The spatial prior Different choices possible for spatial kernel matrix S. Currently used in SPM: Laplacian prior (same as in LORETA)

36. Smoothing Global priorLaplacian Prior Example: application to event-related fMRI data Contrast maps for familiar vs. non-familiar faces, obtained with -smoothing -global spatial prior -Laplacian prior

37. SPM5 graphical user interface

38. Bayesian model selection (BMS) Given competing hypotheses on structure & functional mechanisms of a system, which model is the best? For which model m does p(y|m) become maximal? Which model represents the best balance between model fit and model complexity? Pitt & Miyung (2002), TICS

39. Model evidence: Various approximations, e.g.: -negative free energy -AIC -BIC Penny et al. (2004) NeuroImage Bayesian model selection (BMS) Model comparison via Bayes factor: Bayes’ rules: accounts for both accuracy and complexity of the model allows for inference about structure (generalisability) of the model

40. Example: BMS of dynamic causal models modulation of back- ward or forward connection? additional driving effect of attention on PPC? bilinear or nonlinear modulation of forward connection? V1 V5 stim PPC M2 attention V1 V5 stim PPC M1 attention V1 V5 stim PPC M3 attention V1 V5 stim PPC M4 attention BF = 2966 M2 better than M1 M3 better than M2 BF = 12 M4 better than M3 BF = 23    Stephan et al. (2008) NeuroImage

41. Thank you