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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
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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
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
• normalization: a=2 • marginalization: • conditioning : (Bayes rule) b=5 a=2 Bayesian paradigm (1) : theory of probability Degree of plausibilitydesiderata: - should be represented using real numbers (D1) - should conform with intuition (D2) - should be consistent (D3)
Likelihood: Prior: Bayes rule: generative model m Bayesian paradigm (2) : Likelihood and priors
Model evidence: y = f(x) x model evidence p(y|m) y=f(x) space of all data sets Bayesian paradigm (3) : Model comparison Principle of parsimony : « plurality should not be assumed without necessity » “Occam’s razor”:
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 • subjectivist approach: “informative”priors • objectivist approach: “non-informative”priors order/structure lack of information/entropy Principle of maximum entropy : find the probability distribution function which maximizes the entropy under some constraints (normalization, expectation, …) priors = population behaviour / information available before having observed the data
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
hierarchy causality Hierarchical models (1) : principle •••
prior posterior ••• Hierarchical models (3) : univariate linear hierarchical model
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
MCMC example: Gibbs sampling Sampling methods
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
VB/EM/ReML: find (iteratively) the “variational” posteriorq(θ) which maximizes the free energyF(q) under some mean-field approximation: Variational methods
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
PPM of belonging to… grey matter white matter CSF aMRI segmentation
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
prior variance of GLM coeff prior variance of data noise GLM coeff AR coeff (correlated noise) observations smoothed W (RFT) aMRI ML estimate of W VB estimate of W fMRI time series analysiswith spatial priors
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
state-space formulation: Dynamic causal modelling
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
Homo apriorius Homo pragmaticus Homo frequentistus Homo sapiens Homo bayesianis