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

What is Bayesian Inference ? (From Daniel Wolpert)

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

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

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

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

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

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

General Linear Model Model:

Prior Model: Prior:

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

Priors and likelihood Model: Prior: Likelihood:

Priors and likelihood Model: Prior: Likelihood:

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

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

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

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

SPM Interface

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

ROC curve Sensitivity 1-Specificity

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)

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

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

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

Model Evidence Bayes Rule: normalizing constant Model evidence

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

Bayes factor: Prior Posterior Evidence Model For Equal Model Priors

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

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

p=0.05 Bayesian BF=3 Classical

BF=20 Bayesian BF=3 Classical

p=0.05 BF=20 Bayesian BF=3 Classical

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

Model Evidence Revisited

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

Free Energy Optimisation Initial Point Precisions, a Parameters, q

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

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

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

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

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

log p(yn|m) Gibbs Sampling

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

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

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

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

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

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

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

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