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Wellcome Centre for Neuroimaging, UCL, UK.

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Presentation on theme: "Wellcome Centre for Neuroimaging, UCL, UK."— Presentation transcript:

1 Wellcome Centre for Neuroimaging, UCL, UK.
Bayesian Inference Will Penny Wellcome Centre for Neuroimaging, UCL, UK. SPM for fMRI Course, London, October 21st, 2010

2 What is Bayesian Inference ?
(From Daniel Wolpert)

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

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

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

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

7 Overview Parameter Inference Model Inference Model Estimation
GLMs, 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 Model Inference Model Estimation
GLMs, 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 General Linear Model Model:

10 Prior Model: Prior:

11 Prior Model: Prior: Sample curves from prior (before observing any data) Mean curve

12 Priors and likelihood Model: Prior: Likelihood:

13 Priors and likelihood Model: Prior: Likelihood:

14 Posterior after one observation
Model: Prior: Likelihood: Bayes Rule: Posterior:

15 Posterior after two observations
Model: Prior: Likelihood: Bayes Rule: Posterior:

16 Posterior after eight observations
Model: Prior: Likelihood: Bayes Rule: Posterior:

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

18 SPM Interface

19 Posterior Probability Maps
ML aMRI Smooth Y (RFT) AR coeff (correlated noise) prior precision of AR coeff observations GLM prior precision of GLM coeff Observation noise Bayesian q

20 ROC curve Sensitivity 1-Specificity

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

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

23 Dynamic Causal Models Posterior Density Priors Are Physiological
V1 V5 SPC Posterior Density Priors Are Physiological V5->SPC

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

25 Model Evidence Bayes Rule: normalizing constant Model evidence

26 Bayes factor: Model Evidence Posterior Prior Model, m=j Model, m=i SPC

27 Bayes factor: Prior Posterior Evidence Model For Equal Model Priors

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

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

30 p=0.05 Bayesian BF=3 Classical

31 BF=20 Bayesian BF=3 Classical

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

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

34 Model Evidence Revisited

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

36

37

38 Free Energy Optimisation
Initial Point Precisions, a Parameters, q

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

40 m2 m1 incorrect model (m2) correct model (m1) x1 x2 u1 x3 u2 x1 x2 u1
Figure 2

41 m2 m1 Models from Klaas Stephan MOG LG RVF stim. LVF FG LD LD|RVF
LD|LVF MOG LG RVF stim. LVF FG LD|RVF LD|LVF LD m2 m1 Models from Klaas Stephan

42 Random Effects (RFX) Inference
log p(yn|m)

43 Gibbs Sampling Frequencies, r Stochastic Method Assignments, A
Initial Point Frequencies, r Stochastic Method Assignments, A

44 log p(yn|m) Gibbs Sampling

45 m2 m1 11/12=0.92 MOG LG RVF stim. LVF FG LD LD|RVF LD|LVF MOG LG RVF

46

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

48 Compute log-evidence for each model/subject
PPMs for Models Compute log-evidence for each model/subject model 1 model K subject 1 subject N Log-evidence maps

49

50

51 PPMs for Models Compute log-evidence for each model/subject model 1
model K subject 1 subject N Log-evidence maps BMS maps Probability that model k generated data PPM EPM Rosa et al Neuroimage, 2009

52 Computational fMRI: Harrison et al (in prep)
Long Time Scale Short Time Scale Frontal cortex Primary visual cortex

53 Non-nested versus nested comparison
For detecting model B: Non-nested: Compare model A versus model B Nested: versus model AB Penny et al, HBM,2007

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

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

56

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