Bayesian inference

1. Bayesian inference Lee Harrison York Neuroimaging Centre 23 / 10 / 2009

2. Posterior probability maps (PPMs) Spatial priors on activation extent Bayesian segmentation and normalisation Dynamic Causal Modelling realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

3. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

4. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

5. Recognition Introduction Generation time

6. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

7. Ordinary least squares Ordinary least squares Data

8. Ordinary least squares Bases (explanatory variables) Sum of squared errors Ordinary least squares Data and model fit Bases (explanatory variables) Sum of squared errors

9. Ordinary least squares Ordinary least squares Data and model fit Bases (explanatory variables) Sum of squared errors

10. Ordinary least squares Ordinary least squares Data and model fit Bases (explanatory variables) Sum of squared errors

11. Ordinary least squares Data and model fit Ordinary least squares Over-fitting: model fits noise Inadequate cost function: blind to overly complex models Solution: include uncertainty in model parameters Bases (explanatory variables) Sum of squared errors

12. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

13. Bayesian approach:priors and likelihood Model:

14. Bayesian approach:priors and likelihood Model: Prior:

15. Bayesian approach:priors and likelihood Model: Prior: Sample curves from prior (before observing any data) Mean curve

16. Bayesian approach:priors and likelihood Model: Prior: Likelihood:

17. Bayesian approach:priors and likelihood Model: Prior: Likelihood:

18. Bayesian approach:priors and likelihood Model: Prior: Likelihood:

19. Bayesian approach:posterior Model: Prior: Likelihood: Bayes Rule:

20. Bayesian approach:posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

21. Bayesian approach:posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

22. Bayesian approach:posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

23. Bayesian approach:model selection Bayes Rule: normalizing constant Model evidence: Cost function

24. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

25. recognition space space time Hierarchical models generation

26. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

27. True posterior mean-field approximation iteratively improve Approximate posterior Log-model evidence ‘distance’ btw approx. and true posterior cannot compute as do not know free energy L KL F can compute Maximize  minimize KL Variational methods: approximate inference

28. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

29. degree of smoothness Spatial precision matrix Smooth Y(RFT) prior precision of GLM coeff prior precision of AR coeff aMRI prior precision of data noise GLM coeff AR coeff (correlated noise) ML estimate of W VB estimate of W observations Penny et al 2005 fMRI time series analysis with spatial priors

30. Display only voxels that exceed e.g. 95% activation threshold Probability mass pn PPM (spmP_*.img) Posterior density q(wn) probability of getting an effect, given the data mean: size of effectcovariance: uncertainty fMRI time series analysis with spatial priors:posterior probability maps Mean (Cbeta_*.img) Std dev (SDbeta_*.img)

31. 8 250 200 6 150 4 100 2 50 0 0 fMRI time series analysis with spatial priors:single subject -auditory dataset Active != Rest Active > Rest Overlay of effect sizes at voxels where SPM is 99% sure that the effect size is greater than 2% of the global mean Overlay of 2 statistics: This shows voxels where the activation is different between active and rest conditions, whether positive or negative

32. Log-evidence maps subject 1 model 1 subject N model K Compute log-evidence for each model/subject fMRI time series analysis with spatial priors:group data – Bayesian model selection

33. Log-evidence maps BMS maps subject 1 model 1 subject N PPM model K EPM Probability that model k generated data model k Compute log-evidence for each model/subject fMRI time series analysis with spatial priors:group data – Bayesian model selection Joao et al, 2009

34. (very simple) Computational model of sequence learning

35. (very simple) Computational model of sequence learning

36. (very simple) Computational model of sequence learning

37. Single subject design matrices Onsets only Short-term memory model long-term memory model (IT indices are smoother) onsets Missed trials IT indices: H,h,I,i H=entropy; h=surprise; I=mutual information; i=mutual surprise

38. Group analysis - Bayesian Model Selection maps Onsets only Regions best explained by short-term memory model Regions best explained by long-term memory model Including primary visual cortex anterior cingulate parahippocampus

39. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

40. [nxt] [nxt] [nxp] [pxt] MEG/EEG Source Reconstruction Distributed Source model Inversion (recognition) Forward model (generation) • under-determined system • priors required n : number of sensors p : number of dipoles t : number of time samples Mattout et al, 2006

41. Overview • Modeling uncertainty • Introduction • Ordinary least squares • Bayesian approach • Hierarchical models • Variational methods • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modeling

42. Dynamic Causal Modelling:generative model for fMRI and ERPs Hemodynamicforward model:neural activityBOLD Electric/magnetic forward model:neural activityEEGMEG LFP Neural state equation: fMRI ERPs Neural model: 1 state variable per region bilinear state equation no propagation delays Neural model: 8 state variables per region nonlinear state equation propagation delays inputs

43. attention estimated effective synaptic strengths for best model (m4) 15 0.10 PPC 0.39 0.26 10 1.25 0.26 stim V1 V5 0.13 5 models marginal likelihood 0.46 0 m1 m2 m3 m4 Bayesian Model Selection for fMRI m1 m2 m3 m4 attention attention attention attention PPC PPC PPC PPC stim V1 stim V1 stim V1 stim V1 V5 V5 V5 V5 [Stephan et al., Neuroimage, 2008]

44. Thank-you

47. fMRI study – sequence learning paradigm Design matrix of GLM Harrison et al 2006 Neural Networks