Bayesian Inference and Bayesian Inference and Posterior Probability Maps Posterior Probability Maps

Bayesian Inference and Bayesian Inference and Posterior Probability Maps Posterior Probability Maps

Guillaume FlandinGuillaume Flandin

Wellcome Department of Imaging Neuroscience,Wellcome Department of Imaging Neuroscience,University College London, UKUniversity College London, UK

SPM Course, London, May 2005SPM Course, London, May 2005

Overview

Introduction Bayesian inference

Segmentation and Normalisation Gaussian Prior and Likelihood Posterior Probability Maps (PPMs)

Global shrinkage prior (2nd level) Spatial prior (1st 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

)|)(( 0HYfp

Shortcomings of this approach:

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

Probability of the data, given no activation

)|( Yp

One can never accept the null hypothesis

Correction for multiple comparisons

Given enough data, one can always demonstrate a significant effect at every voxel

)(

)()|()|(

Yp

pYpYp

Baye’s Rule

YY

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

)(

),()|(

Yp

YpYp

)(

),()|(

p

YpYp

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

)(

)()|()|(

Yp

pYpYP

Overview

Introduction Bayesian inference

Segmentation and Normalisation Gaussian Prior and Likelihood Posterior Probability Maps (PPMs)

Global shrinkage prior (2nd level) Spatial prior (1st level)

Bayesian Model Comparison Comparing Dynamic Causal Models (DCMs)

Summary

SPM5

Bayes and Spatial Preprocessing

NormalisationNormalisation

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

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)

Squared distance between parameters and their expected values

(regularisation)

MAP:MAP:

Deformation parametersDeformation parameters

Unlikely deformationUnlikely deformation

Bayes and Spatial Preprocessing

SegmentationSegmentation

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 (2nd level) Spatial prior (1st level)

Bayesian Model Comparison Comparing Dynamic Causal Models (DCMs)

Summary

SPM5

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

p

pmNyp

Overview

Introduction Bayesian inference

Segmentation and Normalisation Gaussian Prior and Likelihood Posterior Probability Maps (PPMs)

Global shrinkage prior (2nd level) Spatial prior (1st level)

Bayesian Model Comparison Comparing Dynamic Causal Models (DCMs)

Summary

SPM5

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

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

Bayesian fMRI: what prior?

)1( XY

General Linear Model:

Shrinkage prior:

),0()( )1( CNp

)2(0 ),0()( CNp

)(p

0

In the absence of evidenceto the contrary, parameters

will shrink to zero

If Cε and Cβ are known, then the posterior is:

YCXCm

CXCXCT

YY

TY

1||

111|

We are looking for the same effect over multiple voxels

Pooled estimation of Cβ over voxels via EM

Bayesian Inference

LikelihoodLikelihood PriorPriorPosteriorPosterior

SPMsSPMsSPMsSPMs

PPMsPPMsPPMsPPMs

u

)(yft

)0|( tp)|( yp

)()|()|( pypyp

Bayesian testBayesian test Classical T-testClassical T-test

Posterior Probability Maps

)|( yp )|( yp

Posterior distribution: probability of getting an effect, given the dataPosterior 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 dataPosterior 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%)

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: variabilitymean: size of effectprecision: variability

Posterior Probability Maps

Mean (Cbeta_*.img)Mean (Cbeta_*.img)

Std dev (SDbeta_*.img)Std dev (SDbeta_*.img)

PPM (spmP_*.img)PPM (spmP_*.img)

Activation threshold Activation threshold

Probability Probability

Posterior probability distribution p( |Y)Posterior probability distribution p( |Y)

)|( yp

SPM and PPM (PET)S

PM

mip

[0, 0

, 0]

<

< <

PPM2.06

rest [2.06]

SPMresults:C:\home\spm\analysis_PET

Height threshold P = 0.95

Extent threshold k = 0 voxels

Design matrix1 4 7 10 13 16 19 22

147

1013161922252831343740434649525560

contrast(s)

4

SP

Mm

ip[0

, 0, 0

]

<

< <

SPM{T39.0

}

rest

SPMresults:C:\home\spm\analysis_PET

Height threshold T = 5.50

Extent threshold k = 0 voxels

Design matrix1 4 7 10 13 16 19 22

147

1013161922252831343740434649525560

contrast(s)

3

PPMs: Show activations greater than a given size

PPMs: Show activations greater than a given size

SPMs: Show voxels with non-zeros activations

SPMs: Show voxels with non-zeros activations

Verbal fluency dataVerbal fluency data

SPM and PPM (fMRI 1st level)

SPMSPMPPMPPMC. Buchel et al, Cerebral Cortex, 1997C. Buchel et al, Cerebral Cortex, 1997

SPM and PPM (fMRI 2nd level)

SPMSPMPPMPPMR. Henson et al, Cerebral Cortex, 2002R. 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

DisadvantagesDisadvantagesAdvantagesAdvantages

■ 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

11stst level level11stst level level

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

1

1

1

1

ji

ij

d

dD 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

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 AlgorithmVariational Bayes Algorithm

Event related fMRI: familiar versus unfamiliar faces

Global priorGlobal prior Spatial PriorSpatial Prior

Smoothing

Convergence & Sensitivity

o Global o Spatialo SmoothingS

ensi

tivity

Iteration Number

F

1-Specificity

ROC curveConvergence

SPM5 Interface

Overview

Introduction Bayesian inference

Segmentation and Normalisation Gaussian Prior and Likelihood Posterior Probability Maps (PPMs)

Global shrinkage prior (2nd level) Spatial prior (1st level)

Bayesian Model Comparison Comparing Dynamic Causal Models (DCMs)

Summary

SPM5

Bayesian Model Comparison

)(

)()|()|(

Yp

mpmYPYmp

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):

AccuracyAccuracy ComplexityComplexity

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

Comparing Dynamic Causal Models

V1

V5

SPC

Motion

Photic

Attention

0.85

0.57-0.02

0.84

0.58

V1

V5

SPC

Motion

Photic

Attention

0.85

0.57 -0.02

1.360.70

0.85

0.23

V1

V5

SPC

Motion

Photic

Attention

0.86

0.56 -0.02

1.42

0.55

0.75

0.89

12

( | 1)

( | 2)

p y mB

p y m

( | ) ( | , ) ( | )p y m p y m p m d Bayesian Evidence:

Bayes factors:

m=1

m=3

m=2

0.70

Attention0.03

Summary

■ Bayesian inference:– There is no null hypothesis

– Allows to incorporate some prior belief

■ Posterior Probability MapsEmpirical Bayes for 2nd level analyses:

Global shrinkage priorVariational 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 (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.