Bayesian Model Comparison

1. SPC V1 V5 SPC V1 V5 Bayesian Model Comparison Will Penny Wellcome Centre for Neuroimaging, UCL, UK. London-Marseille Joint Meeting, Institut de Neurosciences Cognitive de la Mediterranee, Marseille, September 28-29, 2009

2. Overview • Priors, likelihoods and posteriors • Model selection using evidence • Model selection for groups • Comparing model families

3. Overview • Priors, likelihoods and posteriors • Model selection using evidence • Model selection for groups • Comparing model families

4. Bayesian Paradigm:priors and likelihood Model:

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

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

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

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

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

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

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

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

13. Overview • Priors, likelihoods and posteriors • Model selection using evidence • Model selection for groups • Comparing model families

14. Model Selection Bayes Rule: normalizing constant Model evidence: Cost function

15. Prior Posterior Likelihood SPC V1 V5 Second level of Bayesian Inference Parameters: Model, m Parameter Parameter Model Model Evidence Prior Posterior

16. SPC V1 V5 SPC V1 V5 Bayes Factors Model, m=i Model Evidences: Bayes factor: Model, m=j 1 to 3: Weak 3 to 20: Positive 20 to 100: Strong >100: Very Strong

17. Overview • Priors, likelihoods and posteriors • Model selection using evidence • Dynamic Causal Models • Model selection for groups • Comparing model families

18. Single region u1 c u1 a11 z1 u2 z1 z2

19. u1 c a11 z1 a21 z2 a22 Multiple regions u1 u2 z1 z2

20. Modulatory inputs u1 u2 c u1 a11 z1 u2 b21 z1 a21 z2 z2 a22

21. u1 u2 c u1 a11 z1 u2 b21 a12 z1 a21 z2 z2 a22 Reciprocal connections

22. Overview • Priors, likelihoods and posteriors • Model selection using evidence • Dynamic Causal Models • Model selection for groups • Comparing model families

23. u2 u2 x3 x3 x2 x2 x1 x1 u1 u1 incorrect model (m2) correct model (m1) m2 m1 Figure 2

24. LD LD|LVF LD|RVF LD|LVF LD LD RVF stim. LD LVF stim. RVF stim. LD|RVF LVF stim. MOG MOG MOG MOG LG LG LG LG FG FG FG FG m2 m1 Models from Klaas Stephan

25. Random Effects Inference Different subjects can use different models. is the probability that model m is used in the population at large. We wish to make an inference about this.

27. Overview • Priors, likelihoods and posteriors • Model selection using evidence • Model selection for groups • Comparing model families

28. P F DCM of Auditory Word Processing: Data from an fMRI study by Alex Leff and Tom Schofield P: Posterior STS A: Anterior STS F: Inferior Frontal Gyrus A How does processing change for speech versus reversed speech input ? 2^6=64 possible patterns of ‘modulation’. 2^3=8-1=7 possible patterns of input connectivity 7*64=448 possible networks 26*448=11,648 models in group of 26 subjects

29. Input families: Where does the input go ?

30. A F P AF PA PF PAF

31. P P P P F F F F Four of the top 16 models: (b) (a) A A A A (d) (c)

32. Bayesian Model Averaging

33. Same but now for RFX model probs p(m|Y)

34. P F DCM of Auditory Word Processing: Data from an fMRI study by Alex Leff and Tom Schofield P: Posterior STS A: Anterior STS F: Inferior Frontal Gyrus A How does processing change for speech versus reversed speech input ? (1) Input goes to P. (2) Connections from P to F, and P to A, are increased for speech versus reversed speech

35. Summary • First and second levels of Bayesian inference • Model selection for groups • Comparing model families • DCM for EEG-fMRI • Thank-you !