Bayesian inference

1. Bayesian inference Lee Harrison York Neuroimaging Centre 01 / 05 / 2009

2. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

3. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

4. Recognition Introduction Generation time

5. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

6. Ordinary least squares Curve fitting without Bayes Data

7. Curve fitting without Bayes Data Ordinary least squares

8. Bases (explanatory variables) Sum of squared errors Curve fitting without Bayes Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors

9. Curve fitting without Bayes Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors

10. Curve fitting without Bayes Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors

11. Curve fitting without Bayes Data and model fit Ordinary least squares Over-fitting: model fits noise Inadequate cost function: blind to overly complex models Solution: incorporate uncertainty in model parameters Bases (explanatory variables) Sum of squared errors

12. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

13. Bayesian Paradigm:priors and likelihood Model:

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

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

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

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

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

19. Bayesian Paradigm:posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

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

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

22. Bayesian Paradigm:model selection Bayes Rule: normalizing constant Model evidence: Cost function

23. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

24. recognition space space time Hierarchical models generation

25. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

26. True posterior L KL F Difference btw approx. and true posterior But cannot compute as do not know fixed Can compute Maximize  minimize KL Variational methods:approximate inference and iteratively improve to approximate true posterior Initial guess But how?

27. If you assume posterior factorises then F can be maximised by letting where Variational methods:approximate inference complexity accuracy

28. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

29. degree of smoothness Spatial precision matrix smoothed W (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. smoothed W (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

31. 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)

32. 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

33. 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

34. 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 (submitted)

35. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

36. [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

37. Overview • Probabilistic modeling and representation of uncertainty • Introduction • Curve fitting without Bayes • Bayesian paradigm • Hierarchical models • Variational methods (EM, VB) • SPM applications • fMRI time series analysis with spatial priors • EEG source reconstruction • Dynamic causal modelling

38. 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

39. Thank-you