Bayesian models for fMRI data 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) Methods & models for fMRI data analysis in neuroeconomics November 2010

Why do I need to learn 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

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

Problems of classical (frequentist) statistics p-value: probability of getting the observed data in the effect’s absence. If small, reject null hypothesis that there is no effect. Probability of observing the data y, given no effect ( = 0). 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 the effect, given the observed data

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)

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

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

Bayesian statistics new data prior knowledge 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.

Bayes in motion - an animation

Principles of Bayesian inference 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

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

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

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

Bayesian regression: univariate case Normal densities Univariate linear model Relative precision weighting

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

An intuitive example

Still intuitive

Less intuitive

Bayesian (fixed effects) group analysis 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 analyses in SPM8 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

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 Deformation parameters MAP: “Difference” between template and source image Squared distance between parameters and their expected values (regularisation)

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). p (tissue | intensity) p (intensity | tissue) ∙ p (tissue) Ashburner & Friston 2005, NeuroImage

Bayesian fMRI analyses General Linear Model: with What are the priors? 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

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

Posterior Probability Maps (PPMs) Posterior distribution: probability of the effect given the data mean: size of effect precision: variability 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%)

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

2nd level PPMs with global priors 1st level (GLM): 2nd level (shrinkage prior): Basic idea: use the variance of  over voxels as prior variance of  at any particular voxel. 2nd 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) In the absence of evidence to the contrary, parameters will shrink to zero.

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

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

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.

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

PPMs: pros and cons Advantages Disadvantages One can infer that a cause did not elicit a response Inference is independent of search volume SPMs 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

1st 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

The generative spatio-temporal model q1 q2 r1 r2 u1 u2 a b l W A observation noise regression coefficients autoregressive parameters  = spatial precision of parameters  = observation noise precision  = precision of AR coefficients Y Y=XW+E Penny et al. 2005, NeuroImage

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

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

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

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

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

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