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Bayesian Inference

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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

field theory

statistical

inference

normalisation

p <0.05

template

Bayesian segmentation

and normalisation

Smoothness

modelling

realignment

smoothing

general linear model

Gaussian

field theory

statistical

inference

normalisation

p <0.05

template

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

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

- 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

- 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:

Model:

Prior:

Model:

Prior:

Sample curves from prior (before observing any data)

Mean curve

Model:

Prior:

Likelihood:

Model:

Prior:

Likelihood:

Model:

Prior:

Likelihood:

Bayes Rule:

Posterior:

Model:

Prior:

Likelihood:

Bayes Rule:

Posterior:

Model:

Prior:

Likelihood:

Bayes Rule:

Posterior:

- 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

SPM Interface

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

ROC curve

Sensitivity

1-Specificity

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

Mean (Cbeta_*.img)

PPM (spmP_*.img)

Std dev (SDbeta_*.img)

- 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

SPC

V1

V5

Dynamic Causal Models

Posterior Density

Priors

Are

Physiological

V5->SPC

- 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

Bayes Rule:

normalizing constant

Model evidence

SPC

V1

V5

SPC

V1

Model

Model

Evidence

Prior

Posterior

V5

Bayes factor:

Model, m=j

Model, m=i

Model

Model

Evidence

Prior

Posterior

Bayes factor:

For

Equal

Model

Priors

- 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

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

- 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

Initial Point

Precisions, a

Parameters, q

- 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

u2

u2

x3

x3

x2

x2

x1

x1

u1

u1

incorrect model (m2)

correct model (m1)

m2

m1

Figure 2

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

Random Effects (RFX) Inference

log p(yn|m)

Initial Point

Frequencies, r

Stochastic Method

Assignments, A

log p(yn|m)

Gibbs

Sampling

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

- 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

Log-evidence maps

subject 1

model 1

subject N

model K

Compute log-evidence for each model/subject

Log-evidence maps

subject 1

model 1

subject N

model K

Probability that model k generated data

Compute log-evidence for each model/subject

BMS maps

PPM

EPM

Rosa et al Neuroimage, 2009

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:

Compare model A

versus model AB

Penny et al, HBM,2007

Long

Time

Scale

Short

Time Scale

Frontal cortex

Primary visual cortex

- 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