1. Wellcome Centre for Neuroimaging, UCL, UK.
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
statistical
inference
Gaussian
field theory
normalisation
p <0.05
template

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

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

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

7. Overview Parameter Inference Model Inference Model Estimation
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 Model Inference Model Estimation
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 Model Inference Model Estimation
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. Posterior Probability MapsML
aMRI
Smooth Y (RFT)
AR coeff
(correlated noise)
prior precision
of AR coeff
observations
GLM
prior precision
of GLM coeff
Observation
noise
Bayesian q

20. ROC curve
Sensitivity
1-Specificity

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

22. Overview Parameter Inference Model Inference Model Estimation
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. Dynamic Causal Models Posterior Density Priors Are PhysiologicalV1 V5
SPC
Posterior Density
Priors
Are
Physiological
V5->SPC

24. Overview Parameter Inference Model Inference Model Estimation
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. Model Evidence
Bayes Rule:
normalizing constant
Model evidence

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

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

28. Overview Parameter Inference Model Inference Model Estimation
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 Model Inference Model Estimation
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 Model Inference Model Estimation
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. m2 m1 incorrect model (m2) correct model (m1) x1 x2 u1 x3 u2 x1 x2 u1
Figure 2

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

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

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

44. log p(yn|m)
Gibbs
Sampling

45. m2 m1 11/12=0.92 MOG LG RVF stim. LVF FG LD LD|RVF LD|LVF MOG LG RVF

47. Overview Parameter Inference Model Inference Model Estimation
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. Compute log-evidence for each model/subject
PPMs for Models
Compute log-evidence for each model/subject
model 1
model K
subject 1
subject N
Log-evidence maps

51. PPMs for Models Compute log-evidence for each model/subject model 1
model K
subject 1
subject N
Log-evidence maps
BMS maps
Probability that model k generated data
PPM
EPM
Rosa et al Neuroimage, 2009

52. Computational fMRI: Harrison et al (in prep)
Long
Time
Scale
Short
Time Scale
Frontal cortex
Primary visual cortex

53. Non-nested versus nested comparison
For detecting model B:
Non-nested:
Compare model A
versus model B
Nested:
versus model AB
Penny et al, HBM,2007

54. Double Dissociations Long Time Short Scale Time Scale Frontal cortex
Primary visual cortex

55. Summary Parameter Inference Model Inference Model Estimation
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