Bayes for Beginners

Reverend Thomas Bayes (1702-61)

Velia Cardin

Marta Garrido

http://www.fil.ion.ucl.ac.uk/spm/software/spm2/

“In addition to WLS estimators and classical inference, SPM2 also supports Bayesian estimation and inference. In this instance the statistical parametric maps become posterior probability maps Posterior Probability Maps (PPMs), where the posterior probability is a probability of an effect given the data. There is no multiple comparison problem in Bayesian inference and the posterior probabilities do not require adjustment.”

Overview

VeliaBayes vs Frequentist approachAn exampleBayes theorem

Marta Bayesian InferencePosterior Probability Distribution Bayes in SPMSummary

vs

Bayesian statistics

Frequentist Statistics

With a frequentist approach….

The frequentist conclusion is restricted to the data at hand, it doesn’t take into account previous, valuable information.

In general, we want to relate an event (E) to a hypothesis (H) …and the probability of E given H

If the p-value is sufficiently small, you reject the null hypothesis, but…

it doesn’t say anything about the probability of Hi.

We obtain a p-value that is the probability, given a true H0, for the outcome to be more or equally extreme as the observed outcome.

Conclusions depends on previous evidence. Bayesian approach is not data analysis per se, it brings different types of evidence to answer the questions of importance.

In general, we want to relate an event (E) to a hypothesis (H)and the probability of E given H

Given a prior state of knowledge or belief, it tells how to update beliefs based upon observations (current data).

The probability of a H being true is determined.

With a Bayesian approach…

You can compare the probabilities of different H for a same E

A probability distribution of the parameter or hypothesis is obtained

Macaulay Culkin Busted for Drugs!

Our observations….

DREW BARRYMORE REVEALS ALCOHOL AND DRUG PROBLEMS STARTED AGED EIGHT

Feldman , arrested and charged with heroin possession

Corey Haim in a spiral of prescription drug abuse!

Dana Plato died of a drug overdose at age 34

Todd Bridges on suspicion of shooting and stabbing alleged drug dealer in a crack house. ...

We took a random sample of 40 people, 10 of them were young stars, being 3 of them addicted to drugs. From the other 30, just one.

Our hypothesis is: “Young actors have more probability of becoming drug-addicts”

Drug-addicted

young actors

1 29

10

30

3 7

4 36 40

control

YA+

YA-

D+ D-

With a frequentist approach we will test:

Hi:’Conditions A and B have different effects’ Young actors have a different probability of becoming drug addicts than the rest of the people

H0:’There is no difference in the effect of conditions A and B’

This is not what we want to know!!!

…and we have strong believes that young actors have more probability of becoming drug addicts!!!

The statistical test of choice is 2 and Yates’ correction:

2 = 3.33 p=0.07

We can’t reject the null hypothesis, and the information the p is giving us is basically that if we “do this experiment” many times, 7% of the times we will obtain this result if there is no difference between both conditions.

We want to know if

1 (0.025)29 (0.725)

10 (0.25)

30 (0.75)

3 (0.075)7 (0.175)

4 (0.1) 36 (0.9) 40 (1)

p(D+YA+)

p (D+ YA+) = p (D+ and YA+) / p (YA+)

p (D+YA+) p (YA+D+)

Reformulating

This is Bayes’ Theorem !!!

YA+

YA-

D-D+

total

total

p (YA+ D+) = p (D+ and YA+) / p (D+)

p (D+ YA+) = 0.075 / 0.25 = 0.3

p (YA+ D+) = 0.075 / 0.1 = 0.75

0.3 0.75

p (D+ and YA+) = p (YA+ D+) * p (D+)

p (D+ YA-) = p (D+ and YA-) / p (YA-)

p (D+YA+) > p (D+YA-)

p (D+ YA-) = 0.025 / 0.75 = 0.033

p (D+YA+) > p (D+YA-)

0.3 > 0.033

With a Bayesian approach…

p (D+YA-)

p (YA+D+)p (YA+ D+) * p (D+) p (YA+)

p (D+ YA+)

Substituting p (D+ and YA+) on

p (D+ YA+) = p (D+ and YA+) / p (YA+)

It relates the conditional density of a parameter (posterior probability) with its unconditional density (prior, since depends on information present before the experiment).

The likelihood is the probability of the data given the parameter and represents the data now available.

Bayes’ Theorem for a given parameter

p (data) = p (data) p () / p (data)

1/P (data) is basically a normalizing constant

Posterior likelihood x prior

The prior is the probability of the parameter and represents what was thought before seeing the data.

The posterior represents what is thought given both prior information and the data just seen.

In fMRI….

• Classical – ‘What is the

likelihood of getting these data given no activation occurred?’

• Bayesian (SPM2)– ‘What is the chance of

getting these parameters, given these data?

)(

)|()(

)(

),()|(

yp

ypp

yp

ypyp

)|( yp

)|( yp

What you know about the model after the data arrive, , is

what you knew before, , and what the data told you, .

)(p

In order to make probability statements about given y we begin with a model

)|()(),( yppyp

Bayesian Inference

joint prob. distribution

where

)|()()(

)|()()(

yppyp

yppyp

discrete case

continuous caseor

)|()()|( yppyp

likelihoodpriorposterior

Likelihood: p(y|) = (Md, d-1)

Prior: p() = (Mp, p-1)

Posterior: p(y) ∝ p(y|)*p() = (Mpost, post-1)

Mp

p-1

Mpost

post-1

d-1

Md

post = d + p Mpost = d Md + p Mp

post

Posterior Probability Distributionprecision = 1/2

The effects of different precisions

p = d

p < d

p > d

p ≈ 0

Multivariate Distributions

spatial normalization segmentation

and Bayesian inference in…

Posterior Probability Maps (PPM)Dynamic Causal Modelling (DCM)

SPM uses priors for estimation in…

Bayes in SPM

Shrinkage PriorsSmall, variable effect Large, variable effect

Small, consistent effect Large, consistent effect

Thresholding

p(| y) = 0.95

Summary

In Bayesian estimation we…

1. …start with the formulation of a model that we hope is adequate to describe the situation of interest.

2. …observe the data and when the information available changes it is necessary to update the degrees of belief (probability).

3. …evaluate the fit of the model. If necessary we compute predictive distributions for future observations.

priors over the parameters

posterior distributions

new priors over the parameters

Prejudices or scientific judgment?

The selection of a prior is subjective

and arbitrary.

It is reasonable to draw conclusions in the light of

some reason.

Bayesian methods use probability models for quantifying uncertainty in inferences based on statistical data analysis.

• http://www.stat.ucla.edu/history/essay.pdf (Bayes’ original essay!!!)• http://www.cs.toronto.edu/~radford/res-bayes-ex.html• http://www.gatsby.ucl.ac.uk/~zoubin/bayesian.html

• A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin, 2nd ed. Bayesian Data Analysis. Chapman & Hall/CRC.

• Mackay D. Information Theory, Inference and Learning Algorithms. Chapter 37: Bayesian inference and sampling theory. Cambridge University Press, 2003.

• Berry D, Stangl D. Bayesian Methods in Health-Realated Research. In: Bayesian Biostatistics. Berry D and Stangl K (eds). Marcel Dekker Inc, 1996.

• Friston KJ, Penny W, Phillips C, Kiebel S, Hinton G, Ashburner J. Classical and Bayesian inference in neuroimaging: theory. Neuroimage. 2002 Jun;16(2):465-83.

References

Bayes for Beginners

Reverend Thomas Bayes (1702-61)

…good-bayes!!!

“We don’t see what we don’t seek.”

E. M. Forster