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## PowerPoint Slideshow about ' Bayesian Inference' - talon-franklin

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### Bayesian Inference

Overview

Overview

Overview

Overview

Overview

Overview

Will Penny

Wellcome Centre for Neuroimaging, UCL, UK.

SPM for fMRI Course,

London, October 21st, 2010

(From Daniel Wolpert)

and normalisation

realignment

smoothing

general linear model

Gaussian

field theory

statistical

inference

normalisation

p <0.05

template

and normalisation

Smoothness

modelling

realignment

smoothing

general linear model

Gaussian

field theory

statistical

inference

normalisation

p <0.05

template

and normalisation

Smoothness

estimation

Posterior probability

maps (PPMs)

realignment

smoothing

general linear model

Gaussian

field theory

statistical

inference

normalisation

p <0.05

template

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:

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

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 MapsDisplay 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 MapsMean (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

- 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

- 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

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)

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

subject 1

model 1

subject N

model K

Compute log-evidence for each model/subject

PPMs for Modelssubject 1

model 1

subject N

model K

Probability that model k generated data

Compute log-evidence for each model/subject

PPMs for ModelsBMS 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

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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