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Bayesian inference

Understand Bayesian paradigm, priors, and inference methods like Gibbs sampling for neuroimaging. Learn SPM applications in fMRI and EEG. Overview of Bayesian hierarchical models.

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Bayesian inference

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

  2. 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

  3. 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

  4. Introduction

  5. 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

  6. • 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)

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

  8. 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”:

  9. 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

  10. 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

  11. 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

  12. hierarchy causality Hierarchical models (1) : principle •••

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

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

  15. 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

  16. 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

  17. MCMC example: Gibbs sampling Sampling methods

  18. 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

  19. VB/EM/ReML: find (iteratively) the “variational” posteriorq(θ) which maximizes the free energyF(q) under some mean-field approximation: Variational methods

  20. 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

  21. 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

  22. PPM of belonging to… grey matter white matter CSF aMRI segmentation

  23. 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

  24. 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

  25. 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

  26. state-space formulation: Dynamic causal modelling

  27. 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

  28. EEG source reconstruction

  29. Homo apriorius Homo pragmaticus Homo frequentistus Homo sapiens Homo bayesianis

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