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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. Gaussian

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bayesian inference

Bayesian Inference

Will Penny

Wellcome Centre for Neuroimaging, UCL, UK.

SPM for fMRI Course,

London, October 21st, 2010

slide2

What is Bayesian Inference ?

(From Daniel Wolpert)

slide3

Bayesian segmentation

and normalisation

realignment

smoothing

general linear model

Gaussian

field theory

statistical

inference

normalisation

p <0.05

template

slide4

Bayesian segmentation

and normalisation

Smoothness

modelling

realignment

smoothing

general linear model

Gaussian

field theory

statistical

inference

normalisation

p <0.05

template

slide5

Bayesian segmentation

and normalisation

Smoothness

estimation

Posterior probability

maps (PPMs)

realignment

smoothing

general linear model

Gaussian

field theory

statistical

inference

normalisation

p <0.05

template

slide6

Bayesian segmentation

and normalisation

Smoothness

estimation

Posterior probability

maps (PPMs)

Dynamic Causal

Modelling

realignment

smoothing

general linear model

Gaussian

field theory

statistical

inference

normalisation

p <0.05

template

overview
Overview
  • Parameter Inference
    • 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
overview1
Overview
  • Parameter Inference
    • 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
prior
Prior

Model:

Prior:

prior1
Prior

Model:

Prior:

Sample curves from prior (before observing any data)

Mean curve

priors and likelihood
Priors and likelihood

Model:

Prior:

Likelihood:

priors and likelihood1
Priors and likelihood

Model:

Prior:

Likelihood:

posterior after one observation
Posterior after one observation

Model:

Prior:

Likelihood:

Bayes Rule:

Posterior:

posterior after two observations
Posterior after two observations

Model:

Prior:

Likelihood:

Bayes Rule:

Posterior:

posterior after eight observations
Posterior after eight observations

Model:

Prior:

Likelihood:

Bayes Rule:

Posterior:

overview2
Overview
  • Parameter Inference
    • 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
posterior probability maps

q

Smooth Y(RFT)

prior precision

of GLM coeff

prior precision

of AR coeff

aMRI

Observation

noise

GLM

AR coeff

(correlated noise)

ML

Bayesian

observations

Posterior Probability Maps
slide20

ROC curve

Sensitivity

1-Specificity

posterior probability maps1

Display only voxels that exceed e.g. 95%

activation threshold

Probability mass p

Posterior density

probability of getting an effect, given the data

mean: size of effectcovariance: uncertainty

Posterior Probability Maps

Mean (Cbeta_*.img)

PPM (spmP_*.img)

Std dev (SDbeta_*.img)

overview3
Overview
  • Parameter Inference
    • 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
slide23

SPC

V1

V5

Dynamic Causal Models

Posterior Density

Priors

Are

Physiological

V5->SPC

overview4
Overview
  • Parameter Inference
    • 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

Model evidence

slide26

SPC

V1

V5

SPC

V1

Model

Model

Evidence

Prior

Posterior

V5

Bayes factor:

Model, m=j

Model, m=i

slide27

Model

Model

Evidence

Prior

Posterior

Bayes factor:

For

Equal

Model

Priors

overview5
Overview
  • Parameter Inference
    • 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
slide29

Bayes Factors versus p-values

Two sample t-test

Subjects

Conditions

slide30

p=0.05

Bayesian

BF=3

Classical

slide31

BF=20

Bayesian

BF=3

Classical

slide32

p=0.05

BF=20

Bayesian

BF=3

Classical

slide33

p=0.01

p=0.05

BF=20

Bayesian

BF=3

Classical

overview6
Overview
  • Parameter Inference
    • 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
Free Energy Optimisation

Initial Point

Precisions, a

Parameters, q

overview7
Overview
  • Parameter Inference
    • 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
slide40

u2

u2

x3

x3

x2

x2

x1

x1

u1

u1

incorrect model (m2)

correct model (m1)

m2

m1

Figure 2

slide41

LD

LD|LVF

LD|RVF

LD|LVF

LD

LD

RVF

stim.

LD

LVF

stim.

RVF

stim.

LD|RVF

LVF

stim.

MOG

MOG

MOG

MOG

LG

LG

LG

LG

FG

FG

FG

FG

m2

m1

Models from

Klaas Stephan

gibbs sampling
Gibbs Sampling

Initial Point

Frequencies, r

Stochastic Method

Assignments, A

slide44

log p(yn|m)

Gibbs

Sampling

slide45

LD

LD|LVF

LD|RVF

LD|LVF

LD

LD

RVF

stim.

LD

LVF

stim.

RVF

stim.

LD|RVF

LVF

stim.

MOG

MOG

MOG

MOG

LG

LG

LG

LG

FG

FG

FG

FG

m2

m1

11/12=0.92

overview8
Overview
  • Parameter Inference
    • 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
ppms for models

Log-evidence maps

subject 1

model 1

subject N

model K

Compute log-evidence for each model/subject

PPMs for Models
ppms for models1

Log-evidence maps

subject 1

model 1

subject N

model K

Probability that model k generated data

Compute log-evidence for each model/subject

PPMs for Models

BMS maps

PPM

EPM

Rosa et al Neuroimage, 2009

computational fmri harrison et al in prep
Computational fMRI: Harrison et al (in prep)

Long

Time

Scale

Short

Time Scale

Frontal cortex

Primary visual cortex

slide53

Non-nested versus nested comparison

For detecting model B:

Non-nested:

Compare model A

versus model B

Nested:

Compare model A

versus model AB

Penny et al, HBM,2007

double dissociations
Double Dissociations

Long

Time

Scale

Short

Time Scale

Frontal cortex

Primary visual cortex

summary
Summary
  • Parameter Inference
    • 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
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