Bayesian models for fMRI data

1. Bayesian models for fMRI data Klaas Enno Stephan Translational Neuromodeling Unit (TNU)Institute for Biomedical Engineering, University of Zurich & ETH Zurich Laboratory for Social & Neural Systems Research (SNS), University of Zurich WellcomeTrust Centre for Neuroimaging, University College London With many thanks for slides & images to: FIL Methods group, particularly Guillaume Flandin and Jean Daunizeau The Reverend Thomas Bayes (1702-1761) SPM Course Zurich 13-15 February 2013

3. Bayes‘ Theorem Posterior Likelihood Prior Evidence Reverend Thomas Bayes 1702 - 1761 “Bayes‘ Theorem describes, how an ideally rational person processes information." Wikipedia

4. Bayes’ Theorem Given data y and parameters , the joint probability is: Eliminating p(y,) gives Bayes’ rule: Likelihood Prior Posterior Evidence

5. Bayesian inference: an animation

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

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

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

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

10.  Why should I know about Bayesian stats? Because Bayesian principles are fundamental for • statistical inference in general • sophisticated analyses of (neuronal) systems • contemporary theories of brain function

11. Problems of classical (frequentist) statistics p-value: probability of observing data in the effect’s absence • 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

12. posterior distribution inverse problem Generative models: Forward and inverse problems forward problem likelihood  prior

13. Dynamic causal modeling (DCM) fMRI EEG, MEG Model inversion: Estimating neuronal mechanisms from brain activity measures Forward model: Predicting measured activity given a putative neuronal state Friston et al. (2003) NeuroImage

14. The Bayesian brain hypothesis & free-energy principle sensations – predictions Prediction error Change predictions Change sensory input Action Perception Maximizing the evidence (of the brain's generative model) = minimizing the surprise about the data (sensory inputs). Friston et al. 2006, J Physiol Paris

15. IndividualhierarchicalBayesianlearning volatility associations events in the world sensory stimuli Mathys et al. 2011, Front. Hum. Neurosci.

16. Aberrant Bayesian message passing in schizophrenia: abnormal (precision-weighted) prediction errors  abnormal modulation of NMDAR-dependent synaptic plasticity at forward connections of cortical hierarchies Backward & lateral input Forward & lateral Forward recognition effects De-correlating lateral interactions Backward generation effects Lateral interactions mediating priors Stephan et al. 2006, Biol. Psychiatry

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

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

19. Squared distance between parameters and their expected values (regularisation) “Difference” between template and source image Spatial normalisation: Bayesian regularisation Deformations consist of a linear combination of smooth basis functions (3D DCT). Find maximum a posteriori (MAP) estimates: Deformation parameters MAP:

20. Spatial normalisation: overfitting Affine registration. (2 = 472.1) Template image Non-linear registration without regularisation. (2 = 287.3) Non-linear registration using regularisation. (2 = 302.7)

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

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

23. Posterior Probability Maps (PPMs) Posterior distribution:probability of the effect given the data mean: size of effectprecision: variability Posterior probability map: images of the probability that an activation exceeds some specified threshold, given the data y • Two thresholds: • activation threshold : percentage of whole brain mean signal • probability  that voxels must exceed to be displayed (e.g. 95%)

24. 2nd level PPMs with global priors 1st level (GLM): 2nd level (shrinkage prior): Heuristically:use the variance of mean-corrected activity over voxels as prior variance of  at any particular voxel. (1) reflects regionally specific effects  assume that it is zero on average over voxels variance of this prior is implicitly estimated by estimating (2) 0 In the absence of evidence to the contrary, parameters will shrink to zero.

25. 2nd level PPMs with global priors 1st level (GLM): voxel-specific 2nd level (shrinkage prior): global  pooled estimate over voxels Compute Cε and Cvia ReML/EM, and apply the usual rule for computing posterior mean & covariance for Gaussians: Friston & Penny 2003, NeuroImage

26. PPMs vs. SPMs PPMs Posterior Likelihood Prior SPMs Bayesian test: Classical t-test:

27. PPMs and multiple comparisons Friston & Penny (2003): 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. NB: being debated

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

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

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

31. Bayesian model selection (BMS) Model evidence: Gharamani, 2004 p(y|m) y all possible datasets accounts for both accuracy and complexity of the model • Various approximations, e.g.: • negative free energy, AIC, BIC a measure of generalizability McKay 1992, Neural Comput. Penny et al. 2004a, NeuroImage

32. Approximations to the model evidence Maximizing log model evidence = Maximizing model evidence Logarithm is a monotonic function Log model evidence = balance between fit and complexity No. of parameters In SPM2 & SPM5, interface offers 2 approximations: No. of data points Akaike Information Criterion: Bayesian Information Criterion: Penny et al. 2004a, NeuroImage

33. The (negative) free energy approximation • UnderGaussianassumptionsabouttheposterior (Laplace approximation):

34. The complexity term in F • In contrastto AIC & BIC, thecomplexitytermofthe negative freeenergyFaccountsforparameterinterdependencies. • The complexitytermofFishigher • themoreindependentthepriorparameters ( effective DFs) • themoredependenttheposteriorparameters • themoretheposteriormeandeviatesfromthepriormean • NB: Since SPM8, onlyFisusedformodelselection !

35. Bayes factors To compare two models, we could just compare their log evidences. But: the log evidence is just some number – not very intuitive! A more intuitive interpretation of model comparisons is made possible by Bayes factors: positive value, [0;[ Kass & Raftery classification: Kass & Raftery 1995, J. Am. Stat. Assoc.

36. M3 attention M2 better than M1 PPC BF 2966 F = 7.995 stim V1 V5 M4 attention PPC stim V1 V5 BMS in SPM8: an example attention M1 M2 PPC PPC attention stim V1 V5 stim V1 V5 M3 M1 M4 M2 M3 better than M2 BF 12 F = 2.450 M4 better than M3 BF 23 F = 3.144

37. Thank you