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. 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)

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

8. 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:

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 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, … )

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. Hierarchical models (1) : principle hierarchy causality

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

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

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. Sampling methods MCMC example: Gibbs sampling

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

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. aMRI segmentation grey matterPPM of belonging to … CSFwhite matter

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. 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)

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. Dynamic causal modelling state-space formulation:

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