Bayesian Inference and Posterior Probability Maps Guillaume Flandin Wellcome Department of Imaging Neuroscience, University College London, UK SPM Course, London, May 2005

Overview Introduction Bayesian inference  Segmentation and Normalisation  Gaussian Prior and Likelihood  Posterior Probability Maps (PPMs)  Global shrinkage prior (2 nd level)  Spatial prior (1 st level)  Bayesian Model Comparison  Comparing Dynamic Causal Models (DCMs) Summary SPM5

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. Probability of the data, given no activation  One can never accept the null hypothesis  Correction for multiple comparisons  Given enough data, one can always demonstrate a significant effect at every voxel

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

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

Overview Introduction Bayesian inference  Segmentation and Normalisation  Gaussian Prior and Likelihood  Posterior Probability Maps (PPMs)  Global shrinkage prior (2nd level)  Spatial prior (1 st level)  Bayesian Model Comparison  Comparing Dynamic Causal Models (DCMs) Summary SPM5

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) MAP: Deformation parameters Unlikely deformation

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.

Overview Introduction Bayesian inference  Segmentation and Normalisation  Gaussian Prior and Likelihood  Posterior Probability Maps (PPMs)  Global shrinkage prior (2 nd level)  Spatial prior (1 st level)  Bayesian Model Comparison  Comparing Dynamic Causal Models (DCMs) Summary SPM5

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

Overview Introduction Bayesian inference  Segmentation and Normalisation  Gaussian Prior and Likelihood  Posterior Probability Maps (PPMs)  Global shrinkage prior (2 nd level)  Spatial prior (1 st level)  Bayesian Model Comparison  Comparing Dynamic Causal Models (DCMs) Summary SPM5

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 Parameters and hyperparameters at each level can be estimated using EM algorithm

Bayesian fMRI: what prior? General Linear Model: Shrinkage prior: 0 In the absence of evidence to the contrary, parameters will shrink to zero If C ε and C β are known, then the posterior is:  We are looking for the same effect over multiple voxels  Pooled estimation of C β over voxels via EM

Bayesian Inference Likelihood Prior Posterior SPMsSPMs PPMsPPMs Bayesian test Classical T-test

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 )

SPM and PPM (PET) PPMs: Show activations greater than a given size SPMs: Show voxels with non-zeros activations Verbal fluency data

SPM and PPM (fMRI 1 st level) SPM PPM C. Buchel et al, Cerebral Cortex, 1997

SPM and PPM (fMRI 2 nd level) SPM PPM R. Henson et al, Cerebral Cortex, 2002

PPMs: Pros and Cons ■ One can infer a cause DID NOT elicit a response ■ Inference independent of search volume ■ SPMs conflate effect- size and effect-variability ■ One can infer a cause DID NOT elicit a response ■ Inference independent of search volume ■ SPMs conflate effect- size and effect-variability Disadvantages Advantages ■ Use of priors over voxels is computationally demanding ■ Utility of Bayesian approach is yet to be established ■ Use of priors over voxels is computationally demanding ■ Utility of Bayesian approach is yet to be established

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 1 st level

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

Overview Introduction Bayesian inference  Segmentation and Normalisation  Gaussian Prior and Likelihood  Posterior Probability Maps (PPMs)  Global shrinkage prior (2 nd level)  Spatial prior (1 st level)  Bayesian Model Comparison  Comparing Dynamic Causal Models (DCMs) Summary SPM5

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

Comparing Dynamic Causal Models V1 V5 SPC Motion Photic Attention V1 V5 SPC Motion Photic Attention V1V5SPC Motion Photic Attention Bayesian Evidence: Bayes factors: m=1 m=3 m= Attention 0.03

Summary ■Bayesian inference: –There is no null hypothesis –Allows to incorporate some prior belief ■Posterior Probability Maps  Empirical Bayes for 2 nd level analyses: Global shrinkage prior  Variational Bayes for single-subject analyses: Spatial prior on regression and AR coefficients ■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.