DCM

1
Bayesian models for fMRI data
Klaas Enno Stephan Laboratory for Social and
Neural Systems Research Institute for Empirical
Research in Economics University of
Zurich Functional Imaging Laboratory
(FIL) Wellcome Trust Centre for
Neuroimaging University College London
With many thanks for slides images to FIL
Methods group, particularly Guillaume Flandin
The Reverend Thomas Bayes (1702-1761)
Methods models for fMRI data analysis06 May
2009
2
Why do I need to learn about Bayesian stats?

• Because SPM is getting more and more Bayesian
• Segmentation spatial normalisation
• Posterior probability maps (PPMs)
• 1st level specific spatial priors
• 2nd level global spatial priors
• Dynamic Causal Modelling (DCM)
• Bayesian Model Selection (BMS)
• EEG source reconstruction

3
Bayesian segmentation and normalisation
Spatial priors on activation extent
Posterior probability maps (PPMs)
Dynamic Causal Modelling
Image time-series
Statistical parametric map (SPM)
Design matrix
Kernel
Realignment
Smoothing
General linear model
Gaussian field theory
Statistical inference
Normalisation
p lt0.05
Template
Parameter estimates
4
Problems of classical (frequentist) statistics
p-value probability of getting the observed data
in the effects absence. If small, reject null
hypothesis that there is no effect.
Probability of observing the data y, given no
effect (? 0).

• Limitations
• One can never accept the null hypothesis
• Given enough data, one can always demonstrate a
  significant effect
• Correction for multiple comparisons necessary

Solution infer posterior probability of the
effect
Probability of the effect, given the observed data
5
Overview of topics

• Bayes' rule
• Bayesian update rules for Gaussian densities
• Bayesian analyses in SPM
• Segmentation spatial normalisation
• Posterior probability maps (PPMs)
• 1st level specific spatial priors
• 2nd level global spatial priors
• Bayesian Model Selection (BMS)

6
Bayesian statistics
new data
prior knowledge
posterior ? likelihood prior
Bayes theorem allows one to formally incorporate
prior knowledge into computing statistical
probabilities. Priors can be of different
sortsempirical, principled or shrinkage priors.
The posterior probability of the parameters
given the data is an optimal combination of prior
knowledge and new data, weighted by their
relative precision.
7
Bayes in motion - an animation
8
Bayes rule
Given data y and parameters ?, the conditional
probabilities are
Eliminating p(y,?) gives Bayes rule
Likelihood
Prior
Posterior
Evidence
9
Principles of Bayesian inference

• Formulation of a generative model

likelihood p(y?) prior distribution p(?)

• Observation of data

y

• Update of beliefs based upon observations, given
  a prior state of knowledge

10
Posterior mean variance of univariate Gaussians
Likelihood Prior
Posterior
Posterior
Likelihood
Prior
Posterior mean variance-weighted combination
of prior mean and data mean
11
Same thing but expressed as precision weighting
Likelihood prior
Posterior
Posterior
Likelihood
Prior
Relative precision weighting
12
Same thing but explicit hierarchical perspective
Likelihood Prior
Posterior
Posterior
Likelihood
Prior
Relative precision weighting
13
Bayesian GLM univariate case
Normal densities
Univariate linear model
Relative precision weighting
14
Bayesian GLM multivariate case
General Linear Model
Normal densities
?2
One step if Ce is known. Otherwise iterative
estimation with EM.
?1
15
An intuitive example
16
Less intuitive
17
Even less intuitive
18
Bayesian (fixed effects) group analysis
Under Gaussian assumptions this is easy to
compute
Likelihood distributions from different subjects
are independent ? one can use the posterior from
one subject as the prior for the next
group posterior covariance
individual posterior covariances
group posterior mean
individual posterior covariances and means
Todays posterior is tomorrows prior
19
Bayesian analyses in SPM5

• Segmentation spatial normalisation
• Posterior probability maps (PPMs)
• 1st level specific spatial priors
• 2nd level global spatial priors
• Dynamic Causal Modelling (DCM)
• Bayesian Model Selection (BMS)
• EEG source reconstruction

20
Spatial normalisation Bayesian regularisation
Deformations consist of a linear combination of
smooth basis functions ? lowest frequencies of a
3D discrete cosine transform.

• Find maximum a posteriori (MAP) estimates
  simultaneously minimise
• squared difference between template and source
  image
• squared difference between parameters and their
  priors

Deformation parameters
MAP
21
Bayesian segmentation with empirical priors

• Goal for each voxel, compute probability that it
  belongs to a particular tissue type, given its
  intensity
• Likelihood model Intensities are modelled by a
  mixture of Gaussian distributions representing
  different tissue classes (e.g. GM, WM, CSF).
• Priors are obtained from tissue probability maps
  (segmented images of 151 subjects).

p (tissue intensity)?? p (intensity tissue)
p (tissue)
Ashburner Friston 2005, NeuroImage
22
Unified segmentation normalisation

• Circular relationship between segmentation
  normalisation
• Knowing which tissue type a voxel belongs to
  helps normalisation.
• Knowing where a voxel is (in standard space)
  helps segmentation.
• Build a joint generative model
• model how voxel intensities result from mixture
  of tissue type distributions
• model how tissue types of one brain have to be
  spatially deformed to match those of another
  brain
• Using a priori knowledge about the parameters
  adopt Bayesian approach and maximise the
  posterior probability

Ashburner Friston 2005, NeuroImage
23
Bayesian fMRI analyses
General Linear Model
with
What are the priors?

• In classical SPM, no priors ( flat priors)
• Full Bayes priors are predefined on a principled
  or empirical basis
• Empirical Bayes priors are estimated from the
  data, assuming a hierarchical generative model ?
  PPMs in SPM

Parameters of one level priors for distribution
of parameters at lower level
Parameters and hyperparameters at each level can
be estimated using EM
24
Hierarchical models and Empirical Bayes
Parametric Empirical Bayes (PEB)
Hierarchical model
EM PEB ReML
Single-level model
Restricted Maximum Likelihood (ReML)
25
Posterior Probability Maps (PPMs)
Posterior distribution probability of the effect
given the data
mean size of effectprecision variability
Posterior probability map images of the
probability (confidence) that an activation
exceeds some specified threshold??, given the
data y

• Two thresholds
• activation threshold ? percentage of whole brain
  mean signal (physiologically relevant size of
  effect)
• probability ? that voxels must exceed to be
  displayed (e.g. 95)

26
PPMs vs. SPMs
PPMs
Posterior
Likelihood
Prior
SPMs
Bayesian test
Classical t-test
27
2nd level PPMs with global priors
1st level (GLM)
2nd level (shrinkage prior)
Basic idea use the variance of ? over voxels as
prior variance of ? at any particular voxel. 2nd
level ?(2) average effect over voxels, ?(2)
voxel-to-voxel variation. ?(1) reflects
regionally specific effects ? assume that it
sums to zero over all voxels ? shrinkage prior at
the second level ? variance of this prior is
implicitly estimated by estimating ?(2)
0
In the absence of evidence to the contrary,
parameters will shrink to zero.
28
Shrinkage Priors
Small variable effect
Large variable effect
Small but clear effect
Large clear effect
29
2nd level PPMs with global priors
1st level (GLM)
voxel-specific
2nd level (shrinkage prior)
global ? pooled estimate
Once Ce and C? are known, we can apply the usual
rule for computing the posterior mean
covariance

• We are looking for the same effect over multiple
  voxels
• Pooled estimation of C? over voxels

Friston Penny 2003, NeuroImage
30
PPMs and multiple comparisons
No need to correct for multiple
comparisons Thresholding a PPM at 95
confidence in every voxel, the posterior
probability of an activation?? ? is ? 95. At
most, 5 of the voxels identified could have
activations less than ?. Independent of the
search volume, thresholding a PPM thus puts an
upper bound on the false discovery rate.
31
PPMs vs.SPMs
PPMs Show activations greater than a given size
SPMs Show voxels with non-zero activations
32
PPMs pros and cons
Disadvantages
Advantages

• One can infer that a cause did not elicit a
  response
• Inference is independent of search volume
• SPMs conflate effect-size and effect-variability

• Estimating priors over voxels is computationally
  demanding
• Practical benefits are yet to be established
• Thresholds other than zero require justification

33
1st level PPMs with local spatial priors

• Neighbouring voxels often not independent
• Spatial dependencies vary across the brain
• But spatial smoothing in SPM is uniform
• Matched filter theorem SNR maximal when
  smoothing the data with a kernel which matches
  the smoothness of the true signal
• Basic idea estimate regional spatial
  dependencies from the data and use this as a
  prior in a PPM? regionally specific smoothing?
  markedly increased sensitivity

Contrast map
AR(1) map
Penny et al. 2005, NeuroImage
34
The generative spatio-temporal model
q1
q2
r1
r2
u1
u2
a
b
l
A
W
? spatial precision of parameters ?
observation noise precision ? precision of AR
coefficients
Y
YXWE
Penny et al. 2005, NeuroImage
35
The spatial prior
Prior for k-th parameter
Spatial precision determines the amount of
smoothness
Shrinkage prior
Spatial kernel matrix
Different choices possible for spatial kernel
matrix S. Currently used in SPM Laplacian prior
(same as in LORETA)
36
Example application to event-related fMRI data
Smoothing

• Contrast maps for familiar vs. non-familiar
  faces, obtained with
• smoothing
• global spatial prior
• Laplacian prior

Global prior
Laplacian Prior
37
SPM5 graphical user interface
38
Bayesian model selection (BMS)
Given competing hypotheses on structure
functional mechanisms of a system, which model is
the best?
Which model represents thebest balance between
model fit and model complexity?
For which model m does p(ym) become maximal?
39
Bayesian model selection (BMS)
Bayes rules
Model evidence
accounts for both accuracy and complexity of the
model
allows for inference about structure
(generalisability) of the model

• Various approximations, e.g.
• negative free energy
• AIC
• BIC

Model comparison via Bayes factor
Penny et al. (2004) NeuroImage
40
Example BMS of dynamic causal models
attention
M2
M1
?
modulation of back- ward or forward connection?
PPC
PPC
attention
V1
stim
V1
V5
stim
V5
?
additional driving effect of attention on PPC?
?
bilinear or nonlinear modulation of forward
connection?
Stephan et al. (2008) NeuroImage
41
Thank you