Bayesian model selection under
spatial uncertainty for functional
imaging studies
Alexis Roche
CIBM-Siemens, Lausanne, Switzerland
Neurospin, CEA, Paris, France
alexis.roche@gmail.com
6/14/2012
Where's your signal? OHBM'12 workshop
1
Mass univariate model limitations
1. No control over sensitivity
N/S
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Where's your signal? OHBM'12 workshop
2
Mass univariate model limitations
2. Limited localization power
Significant clusters at FWER=5% depending on cluster forming threshold
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Where's your signal? OHBM'12 workshop
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Mass univariate model limitations
3. Specificity reduced by registration errors
Group t-map
without smoothing
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Group t-map
with 6mm smoothing
Where's your signal? OHBM'12 workshop
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Proposed approach
• Bayesian formulation (addresses 1)
• Regional activity is modeled using a predefined parcellation (addresses 2)
• Explicit registration error model (addresses 3)
6/14/2012
Where's your signal? OHBM'12 workshop
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Statistical model
Registration
error fields
Individual
effect maps
𝑢1
𝛽1
fMRI data
𝑌1
𝑢2
Parcel level
group activity
Voxel level
group activity
𝜔
𝜇
𝜇𝑣 = 𝜔 + 𝜉𝑣
𝛽2
𝑌2
𝑢3
𝛽3
𝑌3
𝑢𝑛
𝛽𝑖 = 𝑋𝐺 𝜇 + 𝜁𝑖
𝛽𝑛
𝑌𝑛
𝑌𝑖 = 𝑋𝑖 𝛽𝑖 + 𝜀𝑖 ∘ 𝑢𝑖 −1
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Where's your signal? OHBM'12 workshop
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Inference via belief propagation (BP)
𝑢1
𝑌1
𝛽1
𝑢𝑖
𝜔
6/14/2012
𝜇
𝛽𝑖
Where's your signal? OHBM'12 workshop
𝑌𝑖
7
Approximate BP
𝑢1
𝑌1
𝛽1
𝑢𝑖
𝜔
6/14/2012
𝜇
𝛽𝑖
Where's your signal? OHBM'12 workshop
𝑌𝑖
8
Approximation schemes
• Monte Carlo
– See (Keller et al, Statistica Sinica 08, MICCAI 09)
– Either slow or inaccurate
• Variational
– Approximate messages by factorized Gaussian
distributions
– Amounts to sequential I-projections
– Fast and numerically stable
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Where's your signal? OHBM'12 workshop
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Variational BP step by step
1. Run usual first-level GLM without spatial
smoothing
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Where's your signal? OHBM'12 workshop
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Variational BP step by step
2. Smooth contrast and variance images via
Raw
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2
𝐸(𝛽𝑖 ) = 𝐺 ∗ 𝛽𝑖
𝑉 𝛽𝑖 = 𝐺 ∗ 𝜎𝑖 2 + 𝐺 ∗ 𝛽𝑖 − (𝐺 ∗ 𝛽𝑖 )2
Contrast image
Contrast variance image
Smoothed
Raw
Where's your signal? OHBM'12 workshop
Smoothed
11
Variational BP step by step
2. Smooth contrast and variance images via
2
𝑉 𝛽𝑖 = 𝐺 ∗ 𝜎𝑖 2 + 𝐺 ∗ 𝛽𝑖 − (𝐺 ∗ 𝛽𝑖 )2
𝐸(𝛽𝑖 ) = 𝐺 ∗ 𝛽𝑖
noise
Contrast image
Raw
6/14/2012
spatial uncertainty
Contrast variance image
Smoothed
Raw
Where's your signal? OHBM'12 workshop
Smoothed
12
Variance estimate
BP
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SPM
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Variational BP step by step
3. Estimate voxel level group parameters using
mixed effects variational Bayes algorithm
Group mean
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Group variance
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Variational BP step by step
4. Estimate parcel level group parameters using
mixed effects variational Bayes algorithm
Effect size
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Probability of positive effect
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Method comparison
“Speaker” effect
BP
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Using SPM effect maps,
assuming zero variance
Where's your signal? OHBM'12 workshop
Using SPM effect
and variance maps
16
Method comparison
“Sentence” effect
BP
6/14/2012
Using SPM effect maps,
assuming zero variance
Where's your signal? OHBM'12 workshop
Using SPM effect
and variance maps
17
Conclusion
• Bayesian model selection may mitigate
sensitivity/specificity issues of mass-univariate
inference
• Fast and numerically stable implementation
• Available in NiPy (www.nipy.org),
development version
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