Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course Zurich, February 2008 Bayesian Inference

RealignmentSmoothing Normalisation General linear model Statistical parametric map 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

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

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. Classical approach shortcomings Shortcomings of this approach: Solution: using the probability distribution of the activation given the data.  One can never accept the null hypothesis  Given enough data, one can always demonstrate a significant effect at every voxel Probability of the data, given no activation Probability of the effect, given the observed data  Posterior probability

Baye ’ s Rule  YY Given p(Y), p(  ) and p(Y,  ) Conditional densities are given by Eliminating p(Y,  ) gives Baye’s rule Likelihood Prior Evidence Posterior

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

Multivariate Gaussian

Bayesian Inference Three steps:  Observation of data  Y  Formulation of a generative model  likelihood p(Y|  )  prior distribution p(  )  Update of beliefs based upon observations, given a prior state of knowledge

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 B 12 p(m 1 |Y)Evidence 1 to Weak 3 to Positive 20 to Strong  150  99 Very strong

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

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

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

Bayes and Spatial Preprocessing Segmentation 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. 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. Empirical priors

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

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

Bayesian fMRI General Linear Model: What are the priors? with 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. AR(1)Contrast  Definition of a spatial prior via Gaussian Markov Random Field  Automatically spatially regularisation of Regression coefficients and AR coefficients

The Generative Model  A   Y Y=X β +E where E is an AR(p) General Linear Model with Auto-Regressive error terms (GLM-AR):

Spatial prior Over the regression coefficients: Shrinkage prior Same prior on the AR coefficients. Spatial kernel matrix Spatial precison: determines the amount of smoothness Gaussian Markov Random Field priors 1 on diagonal elements d ii d ij > 0 if voxels i and j are neighbors. 0 elsewhere

Prior, Likelihood and Posterior The prior: The likelihood: The posterior? The posterior over  doesn’t factorise over k or n.  Exact inference is intractable. p(  |Y) ?

Variational Bayes Approximate posteriors that allows for factorisation 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 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 Variational Bayes Algorithm

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

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

SPM5 Interface

Posterior Probability Maps Posterior distribution: probability of getting an effect, given the data 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%) 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

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

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

Example: auditory dataset 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 ■ 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. ■ 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. Disadvantages Advantages ■ Use of priors over voxels is computationally demanding ■ Practical benefits are yet to be established ■ Threshold requires justification ■ Use of priors over voxels is computationally demanding ■ Practical benefits are yet to be established ■ Threshold requires justification

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

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

MEG/EEG Source Reconstruction (2) likelihood prior posterior likelihoodWMN prior minimum norm functional prior smoothness prior 2-level hierarchical model: 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 (2 nd edition), ■ Classical and Bayesian Inference in Neuroimaging: Theory/Applications, Friston et al., NeuroImage, ■ Posterior Probability Maps and SPMs, Friston and Penny, NeuroImage, ■ Variational Bayesian Inference for fMRI time series, Penny et al., NeuroImage, ■ Bayesian fMRI time series analysis with spatial priors, Penny et al., NeuroImage, ■ Comparing Dynamic Causal Models, Penny et al, NeuroImage, 2004.