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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
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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
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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)
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Hierarchical models Parametric Hierarchical Empirical model
Bayes (PEB) Hierarchical model Restricted Maximimum Likelihood (ReML) Single-level model
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Bayes Rule
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Example 2:Univariate model
Likelihood and Prior Posterior Relative Precision Weighting
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Example 2:Multivariate two-level model
Likelihood and Prior Data-determined parameters Assume diagonal precisions Posterior Precisions Assume Shrinkage Prior
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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
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Pooling assumption Decompose error covariance at each voxel, i, into
a voxel specific term, r(i), and voxel-wide terms.
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What’s new in SPM2 ? Corrections for Non-Sphericity
Posterior Probability Maps (PPMs) Haemodynamic modelling Dynamic Causal Modelling (DCM)
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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
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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
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Correlation in fMRI time series
Model errors for each subject as AR(1) + white noise.
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The Interface PEB OLS Parameters Parameters, and REML Hyperparameters
No Priors Shrinkage priors
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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
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Bayesian Inference: Posterior Probability Maps
PPMs Posterior Likelihood Prior SPMs
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SPMs and PPMs PPMs: Show activations of a given size
SPMs: show voxels with non-zero activations
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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
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The Interface Hemodynamic Modelling
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The hemodynamic model
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Hemodynamics
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Inference with MISO models
FUNCTIONAL SEGREGATION: This voxel IS NOT responsive to attention
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The Interface Dynamic Causal Modelling
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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
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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}
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