Guillaume FlandinWellcome Trust Centre for Neuroimaging

University College London

SPM CourseZurich, February 2008

Bayesian Inference

RealignmentRealignment SmoothingSmoothing

NormalisationNormalisation

General linear modelGeneral linear model

Statistical parametric mapStatistical parametric mapImage time-seriesImage time-series

Parameter estimatesParameter estimates

Design matrix

TemplateTemplate

KernelKernel

Gaussian Gaussian field theoryfield theory

p <0.05p <0.05

StatisticalStatisticalinferenceinference

Bayesian segmentationand normalisation

Spatial priorson activation extent

Posterior probabilitymaps (PPMs)

Dynamic CausalModelling

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

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

)|)(( 0HYfp

Shortcomings of this approach:

Solution: using the probability distribution of the activation given the data.

)|( Yp

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

)()()|()|(

YppYpYp

Baye’s Rule

Y

Given p(Y), p() and p(Y,) Conditional densities are given by

)(),()|(

YpYpYp )(

),()|(

pYpYp

Eliminating p(Y,) gives Baye’s rule

Likelihood Prior

Evidence

Posterior

Gaussian Case

Likelihood and Prior

Posterior

)2( m )1(

Relative Precision Weighting

Prior

Likelihood

Posterior

)2()2()1(

)1()1(

y

1

)2()2()1(

1)1(

)1()1(

,

, |

Np

Nyp

)2()2()1()1(

)2()1(

1)1( , |

ppm

ppmNyp

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

)()()|()|(

YppYpYP

Bayesian Model Comparison

)()()|()|(

YpmpmYPYmp

Select the model m with the highest probability given the data:

Model comparison and Baye’s factor:

)|()|(

2

112 mYp

mYpB

mmm dmpmYpmYp )|(),|()|(

Model evidence (marginal likelihood):

Accuracy Complexity

B12 p(m1|Y) Evidence1 to 3 50-75 Weak3 to 20 75-95 Positive

20 to 150 95-99 Strong 150 99 Very strong

Overview

Introduction Bayes’s rule Gaussian case Bayesian Model Comparison

Bayesian inference Bayesian inference aMRI: Segmentation and NormalisationaMRI: Segmentation and Normalisation fMRI: Posterior Probability Maps (PPMs)

Spatial prior (1st level) MEEG: Source reconstruction

Summary

Bayes and Spatial Preprocessing

Normalisation

)(log)|(log)|(log pypyp

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

Templateimage

Affine registration.

(2 = 472.1)

Non-linearregistration

withoutBayes

constraints.(2 = 287.3)

Without Bayesian constraints, the non-linear spatial normalisation can introduce unnecessary warps.

Non-linearregistration

usingBayes.

(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.

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 inferenceBayesian inference aMRI: Segmentation and Normalisation fMRI: Posterior Probability Maps (PPMs)fMRI: Posterior Probability Maps (PPMs)

Spatial prior (1Spatial prior (1stst level) level) MEEG: Source reconstruction

Summary

Bayesian fMRI

XY

General Linear Model:

What are the priors?

),0( CNwith

• 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)

),0()( 11 DNp kk ),0()( 11 DNap pp

General Linear Model with Auto-Regressive error terms (GLM-AR):

t

p

iititt eaXy

1

Spatial prior

11,0 DNp kk

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 D

11

11

ji

ij

ddD

1 on diagonal elements dii

dij > 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) ?

nn

ppppk

kkk

uup

rrpapqqppAp

),|(

),|()|(),|()|(),,,,(

21

2121

n

nnnn aypAYp ),,|(),,|(

Variational Bayes

Approximate posteriors that allows for factorisation

nnnn

pp

kk YqYaqYqYqYqAq )|()|()|()|()|(),,,,(

InitialisationWhile (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 γEnd

Variational Bayes Algorithm

Event related fMRI: familiar versus unfamiliar faces

Global prior Spatial Prior

Smoothing

Convergence & Sensitivity

o Global o Spatialo SmoothingS

ensi

tivity

Iteration Number

F

1-Specificity

ROC curveConvergence

SPM5 Interface

Posterior Probability Maps

)|( yp )|( yp

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

)|( yp

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 effectprecision: variability

Posterior Probability Maps

Mean (Cbeta_*.img)

Std dev (SDbeta_*.img)

PPM (spmP_*.img)

Activation threshold

Probability

Posterior probability distribution p( |Y)

)|( yp

Bayesian Inference

LikelihoodLikelihood PriorPriorPosteriorPosterior

SPMsSPMs

PPMsPPMs

u

)(yft

)0|( tp)|( yp

)()|()|( pypyp

Bayesian test Classical T-test

PPMs: Show activations greater than a given size

SPMs: Show voxels with non-zeros activations

Example: auditory dataset

0

2

4

6

8

0

50

100

150

200

250

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.

DisadvantagesAdvantages

■ 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 inferenceBayesian inference aMRI: Segmentation and Normalisation fMRI: Posterior Probability Maps (PPMs)

Spatial prior (1st level) MEEG: Source reconstructionMEEG: Source reconstruction

Summary

MEG/EEG Source Reconstruction (1)

Inverse procedure

Forward modellingDistributed

Source model Data

KJ EJKY

- under-determined system- priors requiredEKJY

[nxt] [nxp] [nxt][pxt]

n : number of sensorsp : number of dipolest : number of time samples

Bayesian framework

Mattout et al, 2006

MEG/EEG Source Reconstruction (2)

)()|()|( JpJYpYJp

likelihood priorposterior

222/1)( WJKJYCJU eMAP

likelihood WMN prior

WWC Tj 1 jCNJp ,0~)(

minimum norm functional priorsmoothness prior

1EKJY

20 EJ ),CΝ(E e0~1

),CΝ(E p0~2

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 coefficientsDrawbacks:

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.