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

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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


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


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


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

    • 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

Posterior Probability Maps


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

Posterior Probability Maps

Mean (Cbeta_*.img)

PPM (spmP_*.img)

Std dev (SDbeta_*.img)


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


SPC

V1

V5

Dynamic Causal Models

Posterior Density

Priors

Are

Physiological

V5->SPC


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


Bayes Rule:

normalizing constant

Model Evidence

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


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


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

    • 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

    • 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)


Gibbs Sampling

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


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


Log-evidence maps

subject 1

model 1

subject N

model K

Compute log-evidence for each model/subject

PPMs for Models


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)

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


Double Dissociations

Long

Time

Scale

Short

Time Scale

Frontal cortex

Primary visual cortex


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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