Bayesian inference Lee Harrison York Neuroimaging Centre 23 / 10 / 2009

realignmentsmoothing normalisation general linear model template Gaussian field theory p <0.05 statisticalinference Bayesian segmentation and normalisation Bayesian segmentation and normalisation Spatial priors on activation extent Spatial priors on activation extent Dynamic Causal Modelling Dynamic Causal Modelling Posterior probability maps (PPMs) Posterior probability maps (PPMs)

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

Generation Recognition Introduction time

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

Ordinary least squares DataOrdinary least squares

Bases (explanatory variables)Sum of squared errors Data and model fit Bases (explanatory variables)Sum of squared errors Ordinary least squares

Data and model fit Bases (explanatory variables)Sum of squared errors Ordinary least squares

Data and model fit Bases (explanatory variables)Sum of squared errors Ordinary least squares

Data and model fit Bases (explanatory variables)Sum of squared errors Over-fitting: model fits noise Inadequate cost function: blind to overly complex models Solution: include uncertainty in model parameters Ordinary least squares

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

Bayesian approach: priors and likelihood Model:

Bayesian approach: priors and likelihood Model: Prior:

Sample curves from prior (before observing any data) Mean curve Bayesian approach: priors and likelihood Model: Prior:

Bayesian approach: priors and likelihood Model: Prior: Likelihood:

Bayesian approach: priors and likelihood Model: Prior: Likelihood:

Bayesian approach: priors and likelihood Model: Prior: Likelihood:

Bayesian approach: posterior Model: Prior: Likelihood: Bayes Rule:

Bayesian approach: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

Bayesian approach: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

Bayesian approach: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

Bayesian approach: model selection Cost function Bayes Rule: normalizing constant Model evidence:

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

Hierarchical models recognition time space generation space

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

Variational methods: approximate inference True posterior Log-model evidence ‘distance’ btw approx. and true posterior cannot compute as do not know Approximate posterior iteratively improve mean-field approximation can compute Maximize  minimize KL free energy L F KL

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

AR coeff (correlated noise) prior precision of AR coeff VB estimate of W ML estimate of W aMRI Smooth Y (RFT) fMRI time series analysis with spatial priors observations GLM coeff prior precision of GLM coeff prior precision of data noise Penny et al 2005 degree of smoothnessSpatial precision matrix

Mean (Cbeta_*.img) Std dev (SDbeta_*.img) PPM (spmP_*.img) activation threshold Posterior density q(w n ) Probability mass p n fMRI time series analysis with spatial priors: posterior probability maps probability of getting an effect, given the data mean: size of effect covariance: uncertainty Display only voxels that exceed e.g. 95%

fMRI time series analysis with spatial priors: single subject - auditory dataset Active > Rest Active != Rest Overlay of effect sizes at voxels where SPM is 99% sure that the effect size is greater than 2% of the global mean Overlay of  2 statistics: This shows voxels where the activation is different between active and rest conditions, whether positive or negative

fMRI time series analysis with spatial priors: group data – Bayesian model selection Compute log-evidence for each model/subject model 1 model K subject 1 subject N Log-evidence maps

fMRI time series analysis with spatial priors: group data – Bayesian model selection BMS maps PPM EPM model k Joao et al, 2009 Compute log-evidence for each model/subject model 1 model K subject 1 subject N Log-evidence maps Probability that model k generated data

(very simple) Computational model of sequence learning

Onsets only Short-term memory model long-term memory model (IT indices are smoother) onsets Missed trials IT indices: H,h,I,i H=entropy; h=surprise; I=mutual information; i=mutual surprise Single subject design matrices

Onsets only Regions best explained by short- term memory model Regions best explained by long- term memory model Including primary visual cortex anterior cingulate parahippocampus Group analysis - Bayesian Model Selection maps

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

Distributed Source model MEG/EEG Source Reconstruction Forward model (generation) Inversion (recognition) - under-determined system - priors required Mattout et al, 2006 [n x t][n x p] [n x t] [p x t] n : number of sensors p : number of dipoles t : number of time samples

Overview Modeling uncertainty –Introduction –Ordinary least squares –Bayesian approach –Hierarchical models Variational methods SPM applications –fMRI time series analysis with spatial priors –EEG source reconstruction –Dynamic causal modeling

Neural state equation: Electric/magnetic forward model: neural activity  EEG MEG LFP Neural model: 1 state variable per region bilinear state equation no propagation delays Neural model: 8 state variables per region nonlinear state equation propagation delays fMRIERPs inputs Hemodynamic forward model: neural activity  BOLD Dynamic Causal Modelling: generative model for fMRI and ERPs

m2m2 m1m1 m3m3 m4m4 V1 V5 stim PPC attention V1 V5 stim PPC attention V1 V5 stim PPC attention V1 V5 stim PPC attention [Stephan et al., Neuroimage, 2008] m1m1 m2m2 m3m3 m4m V1 V5 stim PPC attention estimated effective synaptic strengths for best model (m 4 ) models marginal likelihood Bayesian Model Selection for fMRI

Thank-you

Design matrix of GLM Harrison et al 2006 Neural Networks fMRI study – sequence learning paradigm