Bayesian Inference

Bayesian Inference Guillaume FlandinWellcome Trust Centre for NeuroimagingUniversity College London SPM Course Zurich, February 2008

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

Overview • Introduction • Bayes’s rule • Gaussian case • Bayesian Model Comparison • Bayesian inference • aMRI: Segmentation and Normalisation • fMRI: Posterior Probability Maps (PPMs) • Spatial prior (1st level) • MEEG: Source reconstruction • Summary

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 Solution: using the probability distribution of the activation given the data. Probability of the effect, given the observed data  Posterior probability

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

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

Multivariate Gaussian

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:

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

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

Bayes and Spatial Preprocessing Affine registration. (2 = 472.1) Template image Without Bayesian constraints, the non-linear spatial normalisation can introduce unnecessary warps. Non-linear registration without Bayes constraints. (2 = 287.3) Non-linear registration using Bayes. (2 = 302.7)

Bayes and Spatial Preprocessing Segmentation Empirical priors • 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.

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

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

Bayesian fMRI with spatial priors 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

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

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.

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.

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

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

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

SPM5 Interface

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%)

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

Bayesian Inference PPMs Posterior Likelihood Prior SPMs Bayesian test Classical T-test PPMs: Show activations greater than a given size SPMs: Show voxels with non-zeros activations

Example: auditory dataset 8 250 200 6 150 4 100 2 50 0 0 Active != Rest Active > Rest Overlay of effect sizes at voxels where SPM is 99% sure that the effect size is greater than 2% of the global mean Overlay of 2 statistics: This shows voxels where the activation is different between active and rest conditions, whether positive or negative

PPMs: Pros and Cons Disadvantages Advantages • One can infer a cause DID NOT elicit a response • SPMs conflate effect-size and effect-variability whereas PPMs allow to make inference on the effect size of interest directly. • Use of priors over voxels is computationally demanding • Practical benefits are yet to be established • Threshold requires justification

MEG/EEG Source Reconstruction (1) [nxt] [nxt] [nxp] [pxt] Inverse procedure Distributed Source model Data Forward modelling • under-determined system • priors required n : number of sensors p : number of dipoles t : number of time samples Bayesian framework Mattout et al, 2006

MEG/EEG Source Reconstruction (2) posterior likelihood prior 2-level hierarchical model: likelihood WMN prior smoothness prior functional prior minimum norm Mattout et al, 2006

Summary • Bayesian inference: • Incorporation of some prior beliefs, • Preprocessing vs. Modeling • Concept of Posterior Probability Maps. • Variational Bayes for single-subject analyses: • Spatial prior on regression and AR coefficients • Drawbacks: • Computation time: MCMC, Variational Bayes. • Bayesian framework also allows: • Bayesian Model Comparison.

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.

Bayesian Inference