1. Probabilistic Modelling of Brain Imaging Data
Will Penny
The Wellcome Department of Imaging Neuroscience, UCL
http//:

2. Overview Multiple levels of Bayesian Inference
2. A model of fMRI time series: The Noise
3. A model of fMRI time series: The Signal

3. First level of Bayesian Inference
We have data, y, and some parameters, b
First level of Inference: What are the best parameters ?
Parameters are of model, M, ….

4. First and Second Levels
The first level again, writing in dependence on M:
Second level of Inference: What’s the best model ?

5. Model Selection We need to compute the Bayesian Evidence:
We can’t always compute it exactly, but we
can approximate it: Log p(y|M) ~ F(M)
Evidence = Accuracy - Complexity

6. Model Averaging Revisiting the first level:
Model-dependent posteriors are weighted according
to the posterior probability of each model

7. Multiple Levels w3 Y w1 w2
Evidence Up w3 Y w1 w2
Posteriors Down

8. Overview Multiple levels of Bayesian Inference
2. A model of fMRI time series: The Noise
3. A model of fMRI time series: The Signal

9. Remove with ICA/PCA – but non-automatic
Noise sources in fMRI
1. Slow drifts due to instrumentation instabilities
2. Subject movement
3. Vasomotor oscillation ~ 0.1 Hz
4. Respiratory activity ~ 0.25 Hz
5. Cardiac activity ~ 1 Hz
Remove with ICA/PCA – but non-automatic

10. fMRI time series model Use a General Linear Model at each voxel:
y = X b + e
where X contains task-related regressors.

11. fMRI time series model b e y X + = Time-series at one spatial location
Putative effects of
experimental
manipulation
Size
of effects
Residuals

12. fMRI time series model Use a General Linear Model at each voxel:
y = X b + e
where X contains task-related regressors.
The errors are modelled as an AR(p) process.
(Parametric spectral estimation)
The order can be selected using Bayesian evidence

13. Synthetic GLM-AR(3) Data

14. Map of AR model order, p
Face
Data
p=0,1,2,3

15. Angiograms

16. Other subjects, a1
Ring of
voxels with
highly correlated
error

17. Other subjects, a1 Unmodelled signal or increased cardiac artifact due
to increased
blood flow?

18. Overview Multiple levels of Bayesian Inference
2. A model of fMRI time series: The Noise
3. A model of fMRI time series: The Signal

19. fMRI time series model Use a General Linear Model y = X b + e
Priors factorise into groups:
p(b) = p(b1) p(b2) p(b3)
Priors in each group may be smoothness
priors or Gaussians

20. Rik’s data Press left key if famous, right key if not
Every face presented twice
Part of larger study looking
at factors influencing
repetition suppresion
Press left key if famous, right key if not
24 Transverse Slices acquired with TR=2s
Time series of 351 images

21. Modelling the Signal
Assumption: Neuronal Event Stream is
Identical to the Experimental Event Stream
Convolve event-stream with basis functions to account for
the HRF

22. FIR models
Size of
signal
Time
after
event

23. Separate smoothness priors for each event type
FIR model
Design matrix
for FIR model with
8 time bins in a
20-second window
Separate smoothness priors for each event type
Q. Is this a good prior ?

24. FIR basis set Left occipital cortex (x=-33, y=-81, z=-24)
FIR model average responses

25. FIR basis set Right fusiform cortex (x=45, y=-60, z=-18)
FIR model average responses

26. RFX-Event model 97 parameters ! But only 24 effective parameters
Responses to each event of type A are randomly distributed
about some typical “type A” response
Design Matrix
97 parameters ! But only 24 effective parameters

27. Non-stationary models
As RFX-event but smoothness priors
Testing for smooth temporal variations statistically …

28. Comparing Types of Models
Evidence
Right Fusiform
Left Occipital
RFX-Event
FIR
NonStat
RFX-Event
FIR
NonStat
Model averaging to get peak post-stimulus response

29. Summary Bayesian inference provides a framework for
model comparison and synthesis
Appropriate for fMRI as we have some prior knowledge
We have focussed on temporal models