Bayesian inference

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

Introduction



1.1 Introduction







3 SPM applications





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:



1.1 Introduction







3 SPM applications





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, …)



1.1 Introduction







3 SPM applications





Hierarchical models (1) : principle

••• hierarchy

causality

Hierarchical models (2) : directed acyclic graphs (DAGs)

Hierarchical models (3) : univariate linear hierarchical model

•••

prior posterior



1.1 Introduction







3 SPM applications





Sampling methods

MCMC example: Gibbs sampling



1.1 Introduction







3 SPM applications





Variational methods

VB/EM/ReML: find (iteratively) the “variational” posterior q(θ)which maximizes the free energy F(q)under some mean-field approximation:



1.1 Introduction







3 SPM applications





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



1.1 Introduction







3 SPM applications





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)



1.1 Introduction







3 SPM applications





Dynamic causal modelling

state-space formulation:



1.1 Introduction







3 SPM applications





EEG source reconstruction

Homoapriorius

Homopragmaticus

Homofrequentistus

Homosapiens

Homobayesianis

y ,y

y

y

Bayesian inference

Documents

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