1. Bayesian Inference in SPM2
Will Penny
K. Friston, J. Ashburner, J.-B. Poline,
R. Henson, S. Kiebel, D. Glaser
Wellcome Department of Imaging Neuroscience,
University College London, UK

2. SPM99 fMRI time-series Kernel Design matrix
Inference with Gaussian field theory
Statistical parametric map (SPM)
Realignment
Smoothing
General linear model
Normalisation
Adjusted regional data
spatial modes and effective connectivity
Template
Parameter estimates

3. What’s new in SPM2 ? Spatial transformation of images Batch Mode
Modelling and Inference
Expectation-Maximisation (EM)
Restricted Maximum
Likelihood (ReML)
Parametric Empirical
Bayes (PEB)

4. Hierarchical models Parametric Hierarchical Empirical model
Bayes (PEB)
Hierarchical
model
Restricted
Maximimum
Likelihood
(ReML)
Single-level
model

5. Bayes Rule

6. Example 2:Univariate model
Likelihood and Prior
Posterior
Relative Precision Weighting

7. Example 2:Multivariate two-level model
Likelihood and Prior
Data-determined parameters
Assume
diagonal
precisions
Posterior
Precisions
Assume
Shrinkage
Prior

8. General Case: Arbitrary Error Covariances
EM algorithm
E-Step ( ) y C X T 1 - = e q h
M-Step r
for i and j { } { Q tr J g i j ij k å + l
Friston, K. et al. (2002), Neuroimage

9. Pooling assumption Decompose error covariance at each voxel, i, into
a voxel specific term, r(i), and voxel-wide terms.

10. What’s new in SPM2 ? Corrections for Non-Sphericity
Posterior Probability Maps (PPMs)
Haemodynamic modelling
Dynamic Causal Modelling (DCM)

11. Non-sphericity
Relax assumption that errors are Independent and Identically Distributed (IID)
Non-independent errors eg. repeated measures within subject
Non-identical errors eg. unequal condition/subject error variances
Correlation in fMRI time series
Allows multiple parameters at 2nd level ie. RFX

12. Single-subject contrasts from Group FFX
PET Verbal Fluency
SPMs,p<0.001
uncorrected
Single-subject contrasts from Group FFX
Non-identical
error variances
Sphericity
Non-sphericity

13. Correlation in fMRI time series
Model errors for each subject
as AR(1) + white noise.

14. The Interface PEB OLS Parameters Parameters, and REML Hyperparameters
No Priors
Shrinkage
priors

15. Bayesian estimation: Two-level model
1st level = within-voxel
Likelihood
Shrinkage Prior
In the absence of evidence
to the contrary parameters
will shrink to zero
2nd level = between-voxels

16. Bayesian Inference: Posterior Probability Maps
PPMs
Posterior
Likelihood
Prior
SPMs

17. SPMs and PPMs PPMs: Show activations of a given size
SPMs: show voxels with
non-zero activations

18. PPMs Advantages Disadvantages Use of Shrinkage One can infer a cause
priors over voxels
is computationally
demanding
Utility of Bayesian
approach is yet
to be established
One can infer a cause
DID NOT elicit a response
SPMs conflate effect-size
and effect-variability
No multiple comparisons
problem (hence no smoothing)
P-values don’t change with
search volume

19. The Interface
Hemodynamic
Modelling

20. The hemodynamic model

21. Hemodynamics

22. Inference with MISO models
FUNCTIONAL SEGREGATION:
This voxel IS NOT responsive
to attention

23. The Interface
Dynamic
Causal
Modelling

24. Extension to a MIMO system
The bilinear model
neuronal
changes
intrinsic
connectivity
induced
response
Input
u(t)
activity
x1(t)
x3(t)
x2(t)
hemodynamics
response
y(t)=(X)
Hemodynamic model

25. Dynamical Causal Models
Functional integration and the modulation of specific pathways V1 V4
BA37
STG
BA39
Cognitive set - u2(t)
{e.g. semantic processing}
Stimuli - u1(t)
{e.g. visual words}