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 Wellcome Trust 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

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 joint probability is: Eliminating p(y,) gives Bayes’ rule: Likelihood Prior Posterior Evidence

Bayesian inference: 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

 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

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

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

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

sensations – predictions The Bayesian brain hypothesis & free-energy principle sensations – predictions Prediction error Change sensory input Change predictions 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

Individual hierarchical Bayesian learning volatility associations events in the world sensory stimuli Mathys et al. 2011, Front. Hum. Neurosci.

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 g: generative model : expectation of approximate recognition density : parameters of generative model (= connection strengths between levels) : hyperparameters (= parameters encoding the uncertainty of the approximate recognition model) : prediction error Forward recognition effects De-correlating lateral interactions Backward generation effects Lateral interactions mediating priors Stephan et al. 2006, Biol. Psychiatry

 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

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

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

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)

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

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

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 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%)

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) In the absence of evidence to the contrary, parameters will shrink to zero.

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

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

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

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 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

Model comparison and selection 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) 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

Approximations to the model evidence Logarithm is a monotonic function Maximizing log model evidence = Maximizing model evidence 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

The (negative) free energy approximation Under Gaussian assumptions about the posterior (Laplace approximation):

The complexity term in F In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies. The complexity term of F is higher the more independent the prior parameters ( effective DFs) the more dependent the posterior parameters the more the posterior mean deviates from the prior mean NB: Since SPM8, only F is used for model selection !

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;[ B12 p(m1|y) Evidence 1 to 3 50-75% weak 3 to 20 75-95% positive 20 to 150 95-99% strong  150  99% Very strong Kass & Raftery classification: Kass & Raftery 1995, J. Am. Stat. Assoc.

BMS in SPM8: an example M1 M2 M3 M4 attention PPC PPC BF 2966 M2 better than M1 attention stim V1 V5 stim V1 V5 M1 M2 M3 M4 V1 V5 stim PPC M3 attention M3 better than M2 BF  12 F = 2.450 Posterior model probability in lower plot is a normalised probability: p(m_i|y) = p(y|m_i)/sum(p(y|m_i)) Note that under flat model priors p(m_i|y) = p(y|m_i) V1 V5 stim PPC M4 attention M4 better than M3 BF  23 F = 3.144

Thank you