bayesian inference jean daunizeau wellcome trust centre for neuroimaging 16 / 05 / 2008
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Bayesian inference
Jean Daunizeau
Wellcome Trust Centre for Neuroimaging
16 / 05 / 2008
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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 uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
Introduction
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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 paradigm (1) : theory 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 paradigm (2) : Likelihood and priors
generative model m
Likelihood:
Prior:
Bayes rule:
Bayesian paradigm (3) : Model 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:
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
Specification of priors
lack of information/entropy
order/structure
priors = population behaviour / information available before having observed the data
• subjectivist approach : “informative” priors• objectivist approach : “non-informative” priors
Principle of maximum entropy :
find the probability distribution functionwhich maximizes the entropy under some constraints (normalization, expectation, …)
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
Hierarchical models (1) : principle
••• hierarchy
causality
Hierarchical models (2) : directed acyclic graphs (DAGs)
Hierarchical models (3) : univariate linear hierarchical model
•••
prior posterior
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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 uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
Variational methods
VB/EM/ReML: find (iteratively) the “variational” posterior q(θ)which maximizes the free energy F(q)under some mean-field approximation:
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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 uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
aMRI segmentation
grey matterPPM of belonging to… CSFwhite matter
1
2
3
212
223
iiy
class variances
class priorfrequencies
ith voxel label
class means
ith voxel value
1
2
3
212
223
iiy
1
2
3
212
223
iiy
class variances
class priorfrequencies
ith voxel label
class means
ith voxel value
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
fMRI time series analysiswith spatial priors
observations
GLM coeff
prior varianceof GLM coeff
prior varianceof data noise
AR coeff(correlated noise)
ML estimate of W VB estimate of W
aMRI smoothed W (RFT)
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
Dynamic causal modelling
state-space formulation:
Overview of the talk
1 Probabilistic modelling and representation of uncertainty
1.1 Introduction
1.2 Bayesian paradigm
1.3 Specification of priors
1.4 Hierarchical models
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
EEG source reconstruction
Homoapriorius
Homopragmaticus
Homofrequentistus
Homosapiens
Homobayesianis
y ,y
y
y
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