j. daunizeau wellcome trust centre for neuroimaging, london, uk institute of empirical research in...
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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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time
( , , )x f x u
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t t
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DCMs and DAGsa note on causality