J. Daunizeau

Wellcome Trust Centre for Neuroimaging, London, UK

Institute of Empirical Research in Economics, Zurich, Switzerland

Bayesian inference

Prior knowledge on the illumination position

Mamassian & Goutcher 2001.

Overview of the talk

1 Probabilistic modelling and representation of uncertainty1.1 Bayesian paradigm

1.2 Hierarchical models

1.3 Frequentist versus Bayesian inference

2 Numerical Bayesian inference methods

2.1 Sampling methods

2.2 Variational methods (EM, VB)

3 SPM applications

3.1 aMRI segmentation

3.2 fMRI time series analysis with spatial priors

3.3 Dynamic causal modelling

3.4 EEG source reconstruction

Overview of the talk

1 Probabilistic modelling and representation of uncertainty1.1 Bayesian paradigm

1.2 Hierarchical models

1.3 Frequentist versus Bayesian inference

2 Numerical Bayesian inference methods

2.1 Sampling methods

2.2 Variational methods (EM, VB)

3 SPM applications

3.1 aMRI segmentation

3.2 fMRI time series analysis with spatial priors

3.3 Dynamic causal modelling

3.4 EEG source reconstruction

Bayesian paradigmtheory of probability

Degree of plausibility desiderata:- should be represented using real numbers (D1)- should conform with intuition (D2)- should be consistent (D3)

a=2b=5

a=2

• normalization:

• marginalization:

• conditioning :(Bayes rule)

Bayesian paradigmLikelihood and priors

generative model m

Likelihood:

Prior:

Bayes rule:

Bayesian paradigmforward and inverse problems

,p y m

forward problem

likelihood

,p y m

inverse problem

posterior distribution

Bayesian paradigmModel comparison

“Occam’s razor” :

Principle of parsimony :« plurality should not be assumed without necessity »

mo

de

l evi

de

nce

p(y

|m)

space of all data sets

y=f(

x)y

= f(

x)

x

Model evidence:

Hierarchical modelsprinciple

••• hierarchy

causality

Hierarchical modelsdirected acyclic graphs (DAGs)

Hierarchical modelsunivariate linear hierarchical model

•••

prior posterior

Frequentist versus Bayesian inference a (quick) note on hypothesis testing

t t y t *

0*P t t H

0p t H

0*P t t H if then reject H0

• estimate parameters (obtain test stat.)

H0: 0• define the null, e.g.:

• apply decision rule, i.e.:

classical SPM

p y

0P H y

0P H y if then accept H0

• invert model (obtain posterior pdf)

H0: 0• define the null, e.g.:

• apply decision rule, i.e.:

Bayesian PPM

Frequentist versus Bayesian inferenceBayesian inference: what about point hypothesis testing?

0p Y H

1p Y H

Yspace of all datasets

• define the null and the alternative hypothesis in terms of priors, e.g.:

0 0

1 1

1 if 0:

0 otherwise

: 0,

H p H

H p H N

0

1

1P H y

P H yif then reject H0

• invert both generative models (obtain both model evidences)

• apply decision rule, i.e.:

y

Overview of the talk

1 Probabilistic modelling and representation of uncertainty1.1 Bayesian paradigm

1.2 Hierarchical models

1.3 Frequentist versus Bayesian inference

2 Numerical Bayesian inference methods

2.1 Sampling methods

2.2 Variational methods (EM, VB)

3 SPM applications

3.1 aMRI segmentation

3.2 fMRI time series analysis with spatial priors

3.3 Dynamic causal modelling

3.4 EEG source reconstruction

Sampling methods

MCMC example: Gibbs sampling

Variational methods

VB/EM/ReML: find (iteratively) the “variational” posterior q(θ)which maximizes the free energy F(q)

under some approximation

q

1 or 2 p

1 or 2y,m

p

1,

2y,m

1

2

Overview of the talk

1 Probabilistic modelling and representation of uncertainty1.1 Bayesian paradigm

1.2 Hierarchical models

1.3 Frequentist versus Bayesian inference

2 Numerical Bayesian inference methods

2.1 Sampling methods

2.2 Variational methods (EM, VB)

3 SPM applications

3.1 aMRI segmentation

3.2 fMRI time series analysis with spatial priors

3.3 Dynamic causal modelling

3.4 EEG source reconstruction

realignmentrealignment smoothingsmoothing

normalisationnormalisation

general linear modelgeneral linear model

templatetemplate

Gaussian Gaussian field theoryfield theory

p <0.05p <0.05

statisticalstatisticalinferenceinference

Bayesian segmentationand normalisation

Bayesian segmentationand normalisation

Spatial priorson activation extent

Spatial priorson activation extent

Dynamic CausalModelling

Dynamic CausalModelling

Posterior probabilitymaps (PPMs)

Posterior probabilitymaps (PPMs)

aMRI segmentation

grey matter CSFwhite matter

…

…

yi ci

k

2

1

1 2 k

class variances

classmeans

ith voxelvalue

ith voxellabel

classfrequencies

[Ashburner et al., Human Brain Function, 2003]

fMRI time series analysiswith spatial priors

fMRI time series

GLM coeff

prior varianceof GLM coeff

prior varianceof data noise

AR coeff(correlated noise)

[Penny et al., Neuroimage, 2004]

ML estimate of W VB estimate of W

aMRI smoothed W (RFT)

PPM

Dynamic causal modellingmodel comparison on network structures

m2m1 m3 m4

V1 V5stim

PPC

attention

V1 V5stim

PPC

attention

V1 V5stim

PPC

attention

V1 V5stim

PPC

attention

[Stephan et al., Neuroimage, 2008]

m1 m2 m3 m4

15

10

5

0

V1 V5stim

PPC

attention

1.25

0.13

0.46

0.39 0.26

0.26

0.10estimated

effective synaptic strengthsfor best model (m4)

models marginal likelihood

ln p y m

m1

m2

diff

eren

ces

in lo

g- m

ode

l evi

denc

es

1 2ln lnp y m p y m

subjects

fixed effect

random effect

assume all subjects correspond to the same model

assume different subjects might correspond to different models

Dynamic causal modellingmodel comparison for group studies

[Stephan et al., Neuroimage, 2009]

EEG source reconstructionfinessing an ill-posed inverse problem

[Mattout et al., Neuroimage, 2006]

I thank you for your attention.

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DCMs and DAGsa note on causality