Bayesian inference

Lee Harrison

York Neuroimaging Centre

01 / 05 / 2009

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

Generation

Recognition

Introduction

time

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

Curve fitting without BayesData Ordinary least squares

yZZZE

ZyZyE

Zxfy

TTols

D

TD

1)(ˆ0

)()(

)(

y

Curve fitting without BayesData Ordinary least squares

y Z

Curve fitting without BayesOrdinary least squares

Z

Bases (explanatory variables) Sum of squared errors

y

Data and model fit

Bases (explanatory variables) Sum of squared errors

Curve fitting without BayesData and model fit Ordinary least squares

Bases (explanatory variables) Sum of squared errors

y Z

Curve fitting without BayesData and model fit Ordinary least squares

Bases (explanatory variables) Sum of squared errors

y Z

Curve fitting without BayesData and model fit

Bases (explanatory variables) Sum of squared errors

y

Over-fitting: model fits noise

Inadequate cost function: blind to overly complex models

Solution: incorporate uncertainty in model parameters

Ordinary least squares

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

Bayesian Paradigm:priors and likelihood

eZy Model:

Z

Bayesian Paradigm:priors and likelihood

1

2

eZy Model:

Prior:

)2/exp(

),0()(

2

122

T

kk INp

Sample curves from prior (before observing any data)

Mean curve

x

Z

Bayesian Paradigm:priors and likelihood

1

2

eZy Model:

Prior:

)2/exp(

),0()(

2

122

T

kk INp

1

2

Bayesian Paradigm:priors and likelihood

1

2

)2/)(exp(

),(),(

),|(),(

21

111

1

111

ii

ii

N

ii

Zy

ZNyp

ypyp

eZy Model:

Prior:

)2/exp(

),0()(

2

122

T

kk INp

Likelihood:

x

Z

Bayesian Paradigm:priors and likelihood

1

2

eZy Model:

Prior:

)2/exp(

),0()(

2

122

T

kk INp

Likelihood:

)2/)(exp(

),(),(

),|(),(

21

111

1

111

ii

ii

N

ii

Zy

ZNyp

ypyp

x

Z

Bayesian Paradigm:priors and likelihood

1

2

eZy Model:

Prior:

)2/exp(

),0()(

2

122

T

kk INp

Likelihood:

)2/)(exp(

),(),(

),|(),(

21

111

1

111

ii

ii

N

ii

Zy

ZNyp

ypyp

x

Z

Bayesian Paradigm: posterior

yCZ

IZZC

CNyp

T

kT

1

1

21

, ,|

x

Z

eZy Model:

Prior:

)2/exp(

),0()(

2

122

T

kk INp

Likelihood:

Bayes Rule:

)|(),|(),( pypyp

Posterior:

N

iiypyp

111 ),|(),(

1

2

Bayesian Paradigm: posterior

1

2

x

Z

yCZ

IZZC

CNyp

T

kT

1

1

21

, ,|

eZy Model:

Prior:

)2/exp(

),0()(

2

122

T

kk INp

Likelihood:

Bayes Rule:

)|(),|(),( pypyp

Posterior:

N

iiypyp

111 ),|(),(

Bayesian Paradigm: posterior

1

2

x

Z

yCZ

IZZC

CNyp

T

kT

1

1

21

, ,|

eZy Model:

Prior:

)2/exp(

),0()(

2

122

T

kk INp

Likelihood:

Bayes Rule:

)|(),|(),( pypyp

Posterior:

N

iiypyp

111 ),|(),(

Bayesian Paradigm:model selection

|log myp

2

olsZy

Cos

t fu

nct

ion

Bayes Rule:

)(

)|(),|(),(

myp

mpmypmyp

normalizing constant

dmpmypmyp )|(),|()(

)()(

)|(log

mcomplexitymaccuracy

myp

Model evidence:

constkmcomplexity

constZymaccuracy

2

21

2

2

1

log)(

)(

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

y

1

2

1

2

3

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

)|,(

3222111 mpmpmpmyp

myp

Hierarchical models

111

2221

1332 ),0(~

eZy

eZ

Ne

recognition

y

time

space

1

space

2

generation

?)(q

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

Variational methods:approximate inference

)(

)|,(),|(

myp

mypmyp

True posterior

Initial guess and iteratively improve to approximate true posterior

KLFL

myp

qq

q

mypq

q

q

myp

mypq

myp

mypmyp

),|(

)(log)(

)(

)|,(log)(

)(

)(

),|(

)|,(log)(

),|(

)|,(log)(log L

F

KL

fixed

Can computeMaximize minimize KL

Difference btw approx. and true posteriorBut cannot compute as do not know

But how?

)()( ii

qq

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

)(

)|,(log)(

)(mpqKLmyp

q

mypqF

q

Variational methods:approximate inference

accuracy complexity

iii dmypqI \\ )|,(log)(

ii

ii

dI

Iq

)(exp

)(exp

If you assume posterior factorises

then F can be maximised by letting

where

)()( ii

qq

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

AR coeff(correlated noise)

prior precisionof AR coeff

A

VB estimate of WML estimate of W

aMRI smoothed W (RFT)

fMRI time series analysis with spatial priors

observations

GLM coeff

prior precisionof GLM coeff

prior precisionof data noise

Y

W

Penny et al 2005

11,0 LNWp

degree of smoothness Spatial precision matrix

AR coeff(correlated noise)

prior precisionof AR coeff

A

VB estimate of WML estimate of W

aMRI smoothed W (RFT)

)( nwq )( mwq

nwmw

fMRI time series analysis with spatial priors

observations

GLM coeff

prior precisionof GLM coeff

prior precisionof data noise

Y

W

Penny et al 2005

Mean (Cbeta_*.img)

Std dev (SDbeta_*.img)

PPM (spmP_*.img)

)( thsWqp

activation threshold

ths

Posterior density q(wn)

Probability mass pn

fMRI time series analysis with spatial priors:posterior probability maps

probability of getting an effect, given the dataprobability of getting an effect, given the data

),()( nnn Nwq mean: size of effectcovariance: uncertainty

thpp

Display only voxels that exceed e.g. 95%Display only voxels that exceed e.g. 95%

fMRI time series analysis with spatial priors:single subject - 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

|log myp

2

olsZy

fMRI time series analysis with spatial priors:group data – Bayesian model selection

)()(log qFmyp

Compute log-evidence for each model/subjectCompute log-evidence for each model/subject

model 1model 1

model Kmodel K

subject 1subject 1

subject Nsubject N

Log-evidence mapsLog-evidence maps

fMRI time series analysis with spatial priors:group data – Bayesian model selection

kr

k

BMS mapsBMS maps

PPMPPM

EPMEPM

model kmodel kJoao et al, 2009 (submitted)

)()(log qFmyp

Compute log-evidence for each model/subjectCompute log-evidence for each model/subject

model 1model 1

model Kmodel K

subject 1subject 1

subject Nsubject N

Log-evidence mapsLog-evidence maps

)( krq

kr

941.0)5.0( krq

Probability that model k generated data

Probability that model k generated data

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

DistributedSource model

K

J

MEG/EEG Source Reconstruction

KJy

Forward model(generation)

Inversion(recognition)

- under-determined system- priors required

Mattout et al, 2006

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

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

Overview

• Probabilistic modeling and representation of uncertainty– Introduction

– Curve fitting without Bayes

– Bayesian paradigm

– Hierarchical models

• Variational methods (EM, VB)

• SPM applications– fMRI time series analysis with spatial priors

– EEG source reconstruction

– Dynamic causal modelling

),,( uxFx Neural state equation:

Electric/magneticforward model:

neural activityEEGMEGLFP

Neural model:1 state variable per regionbilinear state equationno propagation delays

Neural model:8 state variables per region

nonlinear state equationpropagation delays

fMRI ERPs

inputs

Hemodynamicforward model:neural activityBOLD

Dynamic Causal Modelling:generative model for fMRI and ERPs

Thank-you