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Bayesian inference Jean Daunizeau Wellcome Trust Centre for Neuroimaging 16 / 05 / 2008

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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

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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

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Introduction

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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

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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=2 b=5 a=2 normalization: marginalization: conditioning : (Bayes rule)

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Bayesian paradigm (2) : Likelihood and priors generative model m Likelihood: Prior: Bayes rule:

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Bayesian paradigm (3) : Model comparison Occams razor : Principle of parsimony : « plurality should not be assumed without necessity » model evidence p(y|m) space of all data sets y=f(x) x Model evidence:

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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

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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 function which maximizes the entropy under some constraints (normalization, expectation, … )

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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

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Hierarchical models (1) : principle hierarchy causality

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

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Hierarchical models (3) : univariate linear hierarchical model prior posterior

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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

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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

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Sampling methods MCMC example: Gibbs sampling

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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

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

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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

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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

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aMRI segmentation grey matterPPM of belonging to … CSFwhite matter

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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

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fMRI time series analysis with spatial priors observations GLM coeff prior variance of GLM coeff prior variance of data noise AR coeff (correlated noise) ML estimate of W VB estimate of W aMRI smoothed W (RFT)

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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

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Dynamic causal modelling state-space formulation:

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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

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EEG source reconstruction

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Homo apriorius Homo pragmaticus Homo frequentistus Homo sapiens Homo bayesianis

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