1. J. Daunizeau Wellcome Trust Centre for Neuroimaging, London, UK Institute of Empirical Research in Economics, Zurich, Switzerland Bayesian inference

2. Prior knowledge on the illumination position Mamassian & Goutcher 2001.

3. Overview of the talk 1 Probabilistic modelling and representation of uncertainty 1.1 Bayesian paradigm 1.2 Hierarchical models 1.3 Frequentist versus Bayesian inference 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. Overview of the talk 1 Probabilistic modelling and representation of uncertainty 1.1 Bayesian paradigm 1.2 Hierarchical models 1.3 Frequentist versus Bayesian inference 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

5. Bayesian paradigm 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)

6. Bayesian paradigm Likelihood and priors generative model m Likelihood: Prior: Bayes rule:

7. Bayesian paradigm forward and inverse problems forward problem likelihood inverse problem posterior distribution

8. Bayesian paradigm Model comparison “Occam’s 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. Hierarchical models principle hierarchy causality

10. Hierarchical models directed acyclic graphs (DAGs)

11. Hierarchical models univariate linear hierarchical model prior posterior

12. Frequentist versus Bayesian inference a (quick) note on hypothesis testing ifthen reject H0 estimate parameters (obtain test stat.) define the null, e.g.: apply decision rule, i.e.: classical SPM ifthen accept H0 invert model (obtain posterior pdf) define the null, e.g.: apply decision rule, i.e.: Bayesian PPM

13. Frequentist versus Bayesian inference Bayesian inference: what about point hypothesis testing? space of all datasets define the null and the alternative hypothesis in terms of priors, e.g.: ifthen reject H0 invert both generative models (obtain both model evidences) apply decision rule, i.e.:

14. Overview of the talk 1 Probabilistic modelling and representation of uncertainty 1.1 Bayesian paradigm 1.2 Hierarchical models 1.3 Frequentist versus Bayesian inference 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

15. Sampling methods MCMC example: Gibbs sampling

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

17. Overview of the talk 1 Probabilistic modelling and representation of uncertainty 1.1 Bayesian paradigm 1.2 Hierarchical models 1.3 Frequentist versus Bayesian inference 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

18. realignmentsmoothing normalisation general linear model template Gaussian field theory p <0.05 statisticalinference Bayesian segmentation and normalisation Bayesian segmentation and normalisation Spatial priors on activation extent Spatial priors on activation extent Dynamic Causal Modelling Dynamic Causal Modelling Posterior probability maps (PPMs) Posterior probability maps (PPMs)

19. aMRI segmentation grey matterCSFwhite matter … … class variances class means i th voxel value i th voxel label class frequencies [Ashburner et al., Human Brain Function, 2003]

20. fMRI time series analysis with spatial priors fMRI time series GLM coeff prior variance of GLM coeff prior variance of data noise AR coeff (correlated noise) [Penny et al., Neuroimage, 2004] ML estimate of W VB estimate of W aMRI smoothed W (RFT) PPM

21. Dynamic causal modelling model comparison on network structures m2m2 m1m1 m3m3 m4m4 V1 V5 stim PPC attention V1 V5 stim PPC attention V1 V5 stim PPC attention V1 V5 stim PPC attention [Stephan et al., Neuroimage, 2008] m1m1 m2m2 m3m3 m4m4 15 10 5 0 V1 V5 stim PPC attention 1.25 0.13 0.46 0.39 0.26 0.10 estimated effective synaptic strengths for best model (m 4 ) models marginal likelihood

22. m1m1 m2m2 differences in log- model evidences subjects fixed effect random effect assume all subjects correspond to the same model assume different subjects might correspond to different models Dynamic causal modelling model comparison for group studies [Stephan et al., Neuroimage, 2009]

23. EEG source reconstruction finessing an ill-posed inverse problem [Mattout et al., Neuroimage, 2006]

24. I thank you for your attention.

25. 12 3 12 3 12 3 12 3 time u DCMs and DAGs a note on causality