Bayesian Inference

1. Bayesian Inference Will Penny Wellcome Centre for Neuroimaging, UCL, UK. SPM for fMRI Course, London, October 21st, 2010

2. What is Bayesian Inference ? (From Daniel Wolpert)

3. Bayesian segmentation and normalisation realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

4. Bayesian segmentation and normalisation Smoothness modelling realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

5. Bayesian segmentation and normalisation Smoothness estimation Posterior probability maps (PPMs) realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

6. Bayesian segmentation and normalisation Smoothness estimation Posterior probability maps (PPMs) Dynamic Causal Modelling realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

7. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

8. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

9. General Linear Model Model:

10. Prior Model: Prior:

11. Prior Model: Prior: Sample curves from prior (before observing any data) Mean curve

12. Priors and likelihood Model: Prior: Likelihood:

13. Priors and likelihood Model: Prior: Likelihood:

14. Posterior after one observation Model: Prior: Likelihood: Bayes Rule: Posterior:

15. Posterior after two observations Model: Prior: Likelihood: Bayes Rule: Posterior:

16. Posterior after eight observations Model: Prior: Likelihood: Bayes Rule: Posterior:

17. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

18. SPM Interface

19. q Smooth Y(RFT) prior precision of GLM coeff prior precision of AR coeff aMRI Observation noise GLM AR coeff (correlated noise) ML Bayesian observations Posterior Probability Maps

20. ROC curve Sensitivity 1-Specificity

21. Display only voxels that exceed e.g. 95% activation threshold Probability mass p Posterior density probability of getting an effect, given the data mean: size of effectcovariance: uncertainty Posterior Probability Maps Mean (Cbeta_*.img) PPM (spmP_*.img) Std dev (SDbeta_*.img)

22. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

23. SPC V1 V5 Dynamic Causal Models Posterior Density Priors Are Physiological V5->SPC

24. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

25. Bayes Rule: normalizing constant Model Evidence Model evidence

26. SPC V1 V5 SPC V1 Model Model Evidence Prior Posterior V5 Bayes factor: Model, m=j Model, m=i

27. Model Model Evidence Prior Posterior Bayes factor: For Equal Model Priors

28. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

29. Bayes Factors versus p-values Two sample t-test Subjects Conditions

30. p=0.05 Bayesian BF=3 Classical

31. BF=20 Bayesian BF=3 Classical

32. p=0.05 BF=20 Bayesian BF=3 Classical

33. p=0.01 p=0.05 BF=20 Bayesian BF=3 Classical

34. Model Evidence Revisited

35. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

38. Free Energy Optimisation Initial Point Precisions, a Parameters, q

39. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

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

41. 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

42. Random Effects (RFX) Inference log p(yn|m)

43. Gibbs Sampling Initial Point Frequencies, r Stochastic Method Assignments, A

44. log p(yn|m) Gibbs Sampling

45. 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 11/12=0.92

47. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

48. Log-evidence maps subject 1 model 1 subject N model K Compute log-evidence for each model/subject PPMs for Models