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Bayesian models for fMRI data

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1 Bayesian models for fMRI data
Klaas Enno Stephan Translational Neuromodeling Unit (TNU) Institute for Biomedical Engineering, University of Zurich & ETH Zurich Laboratory for Social & Neural Systems Research (SNS), University of Zurich Wellcome Trust Centre for Neuroimaging, University College London With many thanks for slides & images to: FIL Methods group, particularly Guillaume Flandin and Jean Daunizeau The Reverend Thomas Bayes ( ) SPM Course Zurich 13-15 February 2013


3 Bayes‘ Theorem Posterior Likelihood Prior Evidence
Reverend Thomas Bayes “Bayes‘ Theorem describes, how an ideally rational person processes information." Wikipedia

4 Bayes’ Theorem Given data y and parameters , the joint probability is: Eliminating p(y,) gives Bayes’ rule: Likelihood Prior Posterior Evidence

5 Bayesian inference: an animation

6 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

7 Posterior mean & variance of univariate Gaussians
Likelihood & Prior Posterior Posterior: Likelihood Prior Posterior mean = variance-weighted combination of prior mean and data mean

8 Same thing – but expressed as precision weighting
Likelihood & prior Posterior Posterior: Likelihood Prior Relative precision weighting

9 Same thing – but explicit hierarchical perspective
Likelihood & Prior Posterior Posterior Likelihood Prior Relative precision weighting

10  Why should I know about Bayesian stats?
Because Bayesian principles are fundamental for statistical inference in general sophisticated analyses of (neuronal) systems contemporary theories of brain function

11 Problems of classical (frequentist) statistics
p-value: probability of observing data in the effect’s absence 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

12 posterior distribution
Generative models: Forward and inverse problems forward problem likelihood  prior inverse problem posterior distribution

13 Dynamic causal modeling (DCM)
EEG, MEG fMRI Forward model: Predicting measured activity given a putative neuronal state Model inversion: Estimating neuronal mechanisms from brain activity measures Friston et al. (2003) NeuroImage

14 sensations – predictions
The Bayesian brain hypothesis & free-energy principle sensations – predictions Prediction error Change sensory input Change predictions Action Perception Maximizing the evidence (of the brain's generative model) = minimizing the surprise about the data (sensory inputs). Friston et al. 2006, J Physiol Paris

15 Individual hierarchical Bayesian learning
volatility associations events in the world sensory stimuli Mathys et al. 2011, Front. Hum. Neurosci.

16 Aberrant Bayesian message passing in schizophrenia:
abnormal (precision-weighted) prediction errors abnormal modulation of NMDAR-dependent synaptic plasticity at forward connections of cortical hierarchies Backward & lateral input Forward & lateral g: generative model : expectation of approximate recognition density : parameters of generative model (= connection strengths between levels) : hyperparameters (= parameters encoding the uncertainty of the approximate recognition model) : prediction error Forward recognition effects De-correlating lateral interactions Backward generation effects Lateral interactions mediating priors Stephan et al. 2006, Biol. Psychiatry

17  Why should I know 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

18 Bayesian segmentation Posterior probability
and normalisation Spatial priors on activation extent Posterior probability maps (PPMs) Dynamic Causal Modelling Image time-series Statistical parametric map (SPM) Kernel Design matrix Realignment Smoothing General linear model Statistical inference Gaussian field theory Normalisation p <0.05 Template Parameter estimates

19 Spatial normalisation: Bayesian regularisation
Deformations consist of a linear combination of smooth basis functions (3D DCT). Find maximum a posteriori (MAP) estimates: Deformation parameters MAP: “Difference” between template and source image Squared distance between parameters and their expected values (regularisation)

20 Spatial normalisation: overfitting
Affine registration. (2 = 472.1) Template image Non-linear registration without regularisation. (2 = 287.3) Non-linear registration using regularisation. (2 = 302.7)

21 Bayesian segmentation with empirical priors
Goal: for each voxel, compute probability that it belongs to a particular tissue type, given its intensity Likelihood: Intensities are modelled by a mixture of Gaussian distributions representing different tissue classes (e.g. GM, WM, CSF). Priors: obtained from tissue probability maps (segmented images of 151 subjects). p (tissue | intensity) p (intensity | tissue) ∙ p (tissue) Ashburner & Friston 2005, NeuroImage

22 Bayesian fMRI analyses
General Linear Model: with What are the priors? In “classical” SPM, no priors (= “flat” priors) Full Bayes: priors are predefined Empirical Bayes: priors are estimated from the data, assuming a hierarchical generative model Parameters of one level = priors for distribution of parameters at lower level Parameters and hyperparameters at each level can be estimated using EM

23 Posterior Probability Maps (PPMs)
Posterior distribution: probability of the effect given the data mean: size of effect precision: variability Posterior probability map: images of the probability that an activation exceeds some specified threshold, given the data y Two thresholds: activation threshold : percentage of whole brain mean signal probability  that voxels must exceed to be displayed (e.g. 95%)

24 2nd level PPMs with global priors
1st level (GLM): 2nd level (shrinkage prior): Heuristically: use the variance of mean-corrected activity over voxels as prior variance of  at any particular voxel. (1) reflects regionally specific effects  assume that it is zero on average over voxels  variance of this prior is implicitly estimated by estimating (2) In the absence of evidence to the contrary, parameters will shrink to zero.

25 2nd level PPMs with global priors
1st level (GLM): voxel-specific 2nd level (shrinkage prior): global  pooled estimate over voxels Compute Cε and C via ReML/EM, and apply the usual rule for computing posterior mean & covariance for Gaussians: Friston & Penny 2003, NeuroImage

26 PPMs vs. SPMs PPMs Posterior Likelihood Prior SPMs Bayesian test:
Classical t-test:

27 PPMs and multiple comparisons
Friston & Penny (2003): 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. NB: being debated

28 PPMs vs.SPMs PPMs: Show activations greater than a given size
SPMs: Show voxels with non-zero activations

29 PPMs: pros and cons Advantages Disadvantages
One can infer that a cause did not elicit a response Inference is independent of search volume do not 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

30 Model comparison and selection
Given competing hypotheses on structure & functional mechanisms of a system, which model is the best? Pitt & Miyung (2002) TICS Which model represents the best balance between model fit and model complexity? For which model m does p(y|m) become maximal?

31 Bayesian model selection (BMS)
Model evidence: Gharamani, 2004 p(y|m) y all possible datasets accounts for both accuracy and complexity of the model Various approximations, e.g.: negative free energy, AIC, BIC a measure of generalizability McKay 1992, Neural Comput. Penny et al. 2004a, NeuroImage

32 Approximations to the model evidence
Logarithm is a monotonic function Maximizing log model evidence = Maximizing model evidence Log model evidence = balance between fit and complexity No. of parameters In SPM2 & SPM5, interface offers 2 approximations: No. of data points Akaike Information Criterion: Bayesian Information Criterion: Penny et al. 2004a, NeuroImage

33 The (negative) free energy approximation
Under Gaussian assumptions about the posterior (Laplace approximation):

34 The complexity term in F
In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies. The complexity term of F is higher the more independent the prior parameters ( effective DFs) the more dependent the posterior parameters the more the posterior mean deviates from the prior mean NB: Since SPM8, only F is used for model selection !

35 Bayes factors To compare two models, we could just compare their log evidences. But: the log evidence is just some number – not very intuitive! A more intuitive interpretation of model comparisons is made possible by Bayes factors: positive value, [0;[ B12 p(m1|y) Evidence 1 to 3 50-75% weak 3 to 20 75-95% positive 20 to 150 95-99% strong  150  99% Very strong Kass & Raftery classification: Kass & Raftery 1995, J. Am. Stat. Assoc.

36 BMS in SPM8: an example M1 M2 M3 M4 attention PPC PPC BF 2966
M2 better than M1 attention stim V1 V5 stim V1 V5 M1 M2 M3 M4 V1 V5 stim PPC M3 attention M3 better than M2 BF  12 F = 2.450 Posterior model probability in lower plot is a normalised probability: p(m_i|y) = p(y|m_i)/sum(p(y|m_i)) Note that under flat model priors p(m_i|y) = p(y|m_i) V1 V5 stim PPC M4 attention M4 better than M3 BF  23 F = 3.144

37 Thank you

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