Wellcome Trust Centre for Neuroimaging, UCL, UK.

Transcription

1 Bayesian Inference Will Penny Wellcome Trust Centre for Neuroimaging, UCL, UK. SPM Course, Virginia Tech, January 2012

2 What is Bayesian Inference? (From Daniel Wolpert)

3 Bayesian segmentation and normalisation realignment smoothing general linear model normalisation statistical inference p <0.05 Gaussian field theory template

4 Bayesian segmentation and normalisation Smoothness modelling realignment smoothing general linear model normalisation statistical inference p <0.05 Gaussian field theory template

5 Bayesian segmentation and normalisation Smoothness estimation Posterior probability maps (PPMs) realignment smoothing general linear model normalisation statistical inference p <0.05 Gaussian field theory template

6 Bayesian segmentation and normalisation Smoothness estimation Posterior probability maps (PPMs) Dynamic Causal Modelling realignment smoothing general linear model normalisation statistical inference p <0.05 Gaussian field theory template

7 Overview Parameter Inference PPMs, DCMs Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference

8 Overview Parameter Inference PPMs, DCMs Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference

9 SPM Interface

10 Posterior Probability Maps Y X p 1 1 N 0, L 2 amri Smooth Y (RFT) Observation noise prior precision of GLM coeff prior precision of AR coeff GLM A AR coeff (correlated noise) ML Bayesian Y observations

11 ROC curve Sensitivity 1-Specificity

12 Posterior Probability Maps activation threshold s th Probability mass p Display only voxels that exceed e.g. 95% p p th Mean (Cbeta_*.img) Posterior density PPM (spmp_*.img) probability of getting an effect, given the data Std dev (SDbeta_*.img) q( n) N( n, n) mean: size of effect covariance: uncertainty

13 Overview Parameter Inference PPMs, DCMs Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference

14 Dynamic Causal Models SPC Posterior Density V1 V5 Priors are physiological or encourage stable dynamics V5->SPC

15 Parameter Inference GLMs, PPMs, DCMs Overview Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference

16 Model Evidence Bayes Rule: ) ( ) ( ), ( ), ( y m p m p m y p m y p normalizing constant d m p m y p y m p ) ( ), ( ) ( Model evidence

17 Model Posterior Evidence p( y m) p( m) pm ( y) p( y) Model Prior B ij Bayes factor: p( y m i) p( y m j) Model, m=i Model, m=j SPC SPC V1 V1 V5 V5

18 Model Posterior Evidence p( y m) p( m) pm ( y) p( y) Model Prior B ij Bayes factor: p( y m i) p( y m j) For Equal Model Priors

19 Overview Parameter Inference PPMs, DCMs Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference

20 Bayes Factors versus p-values Two sample t-test Subjects Conditions

21 p=0.05 Bayesian BF=3 Classical

22 Bayesian BF=20 BF=3 Classical

23 p=0.05 Bayesian BF=20 BF=3 Classical

24 p=0.01 p=0.05 Bayesian BF=20 BF=3 Classical

25 Parameter Inference GLMs, PPMs, DCMs Overview Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference

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28 Initial Point Free Energy Optimisation Precisions, Parameters,

29 Parameter Inference GLMs, PPMs, DCMs Overview Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference

30 incorrect model (m 2 ) correct model (m 1 ) u 2 u 2 x 3 x 3 u 1 x 1 x 2 u 1 x 1 x 2 m 2 m 1 Sim ulated data sets Log model evidence differences Figure 2

31 LD LD LVF MOG FG FG MOG MOG FG FG MOG LD RVF LD LVF LD LD LG LG LG LG RVF stim. LD LVF stim. RVF stim. LD RVF LVF stim. Subjects m 2 m 1 Models from Klaas Stephan Log model evidence differences

32 Random Effects (RFX) Inference 0.8 A log p(y a) n m) r Subjects Models Models

33 Initial Point Gibbs Sampling p( r A, y) Frequencies, r Stochastic Method Assignments, A p( A r, Y )

34 log p(y a) log p(y n m) ) ( ] log ) ( exp[log ' ' n n m nm nm nm m n nm g Mult a u u g r m y p u ) ( 0 Dir r a n nm m m ), ( Y r A p ), ( y A r p Gibbs Sampling

35 LD LD LVF MOG FG FG MOG MOG FG FG MOG LD RVF LD LVF LD LD LG LG LG LG RVF stim. LD LVF stim. RVF stim. LD RVF LVF stim. Subjects m 2 m 1 11/12= Log model evidence differences

36 5 p(r 1 >0.5 y) = p(r 1 y) r r 1

37 Parameter Inference GLMs, PPMs, DCMs Overview Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference

38 PPMs for Models log p( y m) F( q) Log-evidence maps model 1 subject 1 subject N model K Compute log-evidence for each model/subject

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41 PPMs for Models log p( y m) F( q) Log-evidence maps BMS maps subject 1 model 1 subject N q( 0.5) r k r k PPM q( r k ) model K Compute log-evidence for each model/subject r k k EPM Probability that model k generated data Rosa et al Neuroimage, 2009

42 Computational fmri: Harrison et al (Frontiers 2010) Short Time Scale Long Time Scale Primary visual cortex Frontal cortex

43 Double Dissociations Short Time Scale Long Time Scale Primary visual cortex Frontal cortex

44 Parameter Inference GLMs, PPMs, DCMs Summary Model Inference Model Evidence, Bayes factors (cf. p-values) Model Estimation Variational Bayes Groups of subjects RFX model inference, PPM model inference