j. daunizeau institute of empirical research in economics, zurich, switzerland brain and spine...
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J. Daunizeau
Institute of Empirical Research in Economics, Zurich, Switzerland
Brain and Spine Institute, Paris, France
Bayesian inference
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 (ReML, EM, VB)
3 SPM applications
3.1 aMRI segmentation
3.2 Decoding of brain images
3.3 Model-based fMRI analysis (with spatial priors)
3.4 Dynamic causal modelling
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 (ReML, EM, VB)
3 SPM applications
3.1 aMRI segmentation
3.2 Decoding of brain images
3.3 Model-based fMRI analysis (with spatial priors)
3.4 Dynamic causal modelling
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 paradigmprobability theory: basics
Bayesian paradigmderiving the likelihood function
- Model of data with unknown parameters:
y f e.g., GLM: f X
- But data is noisy: y f
- Assume noise/residuals is ‘small’:
22
1exp
2p
4 0.05P
→ Distribution of data, given fixed parameters:
22
1exp
2p y y f
f
Likelihood:
Prior:
Bayes rule:
Bayesian paradigmlikelihood, priors and the model evidence
generative model m
Bayesian paradigmforward and inverse problems
,p y m
forward problem
likelihood
,p y m
inverse problem
posterior distribution
Principle of parsimony :« plurality should not be assumed without necessity »
y=f(
x)y
= f(
x)
x
“Occam’s razor” :
mo
de
l evi
de
nce
p(y
|m)
space of all data sets
Model evidence:
Bayesian paradigmmodel comparison
••• hierarchy
causality
Hierarchical modelsprinciple
Hierarchical modelsdirected acyclic graphs (DAGs)
•••
prior densities posterior densities
Hierarchical modelsunivariate linear hierarchical model
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 inferencea (quick) note on hypothesis testing
Y
• 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• apply decision rule, i.e.:
y
Frequentist versus Bayesian inferencewhat about bilateral tests?
1p Y H
0p Y H
space of all datasets
• Savage-Dickey ratios (nested models, i.i.d. priors):
10 1
1
0 ,
0
p y Hp y H p y H
p H
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 (ReML, EM, VB)
3 SPM applications
3.1 aMRI segmentation
3.2 Decoding of brain images
3.3 Model-based fMRI analysis (with spatial priors)
3.4 Dynamic causal modelling
Sampling methodsMCMC example: Gibbs sampling
Variational methodsVB / EM / ReML
→ VB : maximize the free energy F(q) w.r.t. the “variational” posterior q(θ) under some (e.g., mean field, Laplace) approximation
1 or 2q
1 or 2 ,p y m
1 2, ,p y 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 (ReML, EM, VB)
3 SPM applications
3.1 aMRI segmentation
3.2 Decoding of brain images
3.3 Model-based fMRI analysis (with spatial priors)
3.4 Dynamic causal modelling
realignmentrealignment smoothingsmoothing
normalisationnormalisation
general linear modelgeneral linear model
templatetemplate
Gaussian Gaussian field theoryfield theory
p <0.05p <0.05
statisticalstatisticalinferenceinference
segmentationand normalisation
segmentationand normalisation
dynamic causalmodelling
dynamic causalmodelling
posterior probabilitymaps (PPMs)
posterior probabilitymaps (PPMs)
multivariatedecoding
multivariatedecoding
grey matter CSFwhite matter
…
…
yi ci
k
2
1
1 2 k
class variances
classmeans
ith voxelvalue
ith voxellabel
classfrequencies
aMRI segmentationmixture of Gaussians (MoG) model
Decoding of brain imagesrecognizing brain states from fMRI
+
fixation cross
>>
paceresponse
log-evidence of X-Y sparse mappings:effect of lateralization
log-evidence of X-Y bilateral mappings:effect of spatial deployment
fMRI time series analysisspatial priors and model comparison
PPM: regions best explainedby short-term memory model
PPM: regions best explained by long-term memory model
fMRI time series
GLM coeff
prior varianceof GLM coeff
prior varianceof data noise
AR coeff(correlated noise)
short-term memorydesign matrix (X)
long-term memorydesign matrix (X)
m2m1 m3 m4
V1 V5stim
PPC
attention
V1 V5stim
PPC
attention
V1 V5stim
PPC
attention
V1 V5stim
PPC
attention
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
Dynamic Causal Modellingnetwork structure identification
1 2
31 2
31 2
3
1 2
3
time
( , , )x f x u
32
21
13
0t
t t
t t
tu
13u 3
u
DCMs and DAGsa note on causality
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
I thank you for your attention.
A note on statistical significancelessons from the Neyman-Pearson lemma
• Neyman-Pearson lemma: the likelihood ratio (or Bayes factor) test
1
0
p y Hu
p y H
is the most powerful test of size to test the null. 0p u H
MVB (Bayes factor) u=1.09, power=56%
CCA (F-statistics)F=2.20, power=20%
error I rate
1 -
erro
r II
rat
e
ROC analysis
• what is the threshold u, above which the Bayes factor test yields a error I rate of 5%?
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