METHODS DUMMIES

BAYES FOR BEGINNERS

Any given Monday at 12.31 pm

“I’m sure this makes sense, but you lost me about here…”

Bayes for Beginners

• What can Bayes do for you?

• Conditional Probability

• What is Bayes Theorem?

• Bayes in SPM2

What can Bayes do for you?

• Problems with classical statistics approach– All inferences relate to disproving the null

hypothesis

– Never fully reject H0, only say that the effect you see is unlikely to occur by chance

– Corrections for multiple comparisons– Very small effects can be declared significant

with enough data

• Bayesian Inference offers a solution

What can Bayes’ do for you?

• Classical– ‘What is the likelihood of getting these data

given no activation occurred (?’– p(y|)

• Bayesian– ‘What is the chance of getting these

parameters, given these data?’– p(|y)

• p(|y) ≠p(y|)

Conditional Probability

• Last year you were at a conference in Japan• You happen to notice that rather a lot of the professors smoke• At one of socials you met someone at the bar & had a few drinks• The next morning you wake up & it dawns on you that you told the person you were

talking to in the bar something rather indiscrete about you supervisor• You remember that the person you were talking kept stealing your cigarettes, and you

are start to worry that they might have been a professor (and therefore a friend of your supervisor)

• You decide to do a calculation to work out what the chances are that the person you were talking to is a professor, given that you know they are a smoker.

• You phone the hotel reception & they give you the following information:– 100 delegates

– 40 requested non-smoking rooms

– 10 professors

– 4 requested non-smoking rooms

p(P) = 0.1

p(P’) = 0.9

Given that the person you were talking to last night was a smoker, what is the probability of them being a professor?

p(S|P) = 0.6

p(S’|P) = 0.4

p(S|P’) = 0.6

p(S’|P’) = 0.4

p(P|S) = p(P and S) p(S)

p(S) = p(S and P) or p(S and P’)

‘AND’ = multiply

‘OR’ = add

p(S and P) = p(S|P)*p(P) = 0.06

p(S and P’) = p(S|P’)*p(P’)

p(P|S) = p(S|P)*p(P) p(S) = 0.06/0.6 p(P|S) = 0.1 p(S) = 0.6*0.1 + 0.6*0.9

= 0.6

100 delegates40 non-smokers10 professors4 non-smokers

The following night you are introduced to a professor who you would very much like to work for after you have finished your PhD. You want to make a

good impression. Given that this person is a professor, what are the chances that they are also a smoker, in which case offering them a cigarette

won’t harm your career prospects?

p(P) = 0.1

p(P’) = 0.9

p(S|P) = 0.6

p(S’|P) = 0.4

p(S|P’) = 0.6

p(S’|P’) = 0.4

p(P|S) ≠ p(S|P)

p(S|P) = p(S and P) p(P)

= 0.6 * 0.1 / 0.1 = 0.6

p(P|S) = p(S|P)*p(P) p(S) = 0.06/0.6 = 0.1

What is Bayes Theorem?

• p(P|S) = p(S|P)*p(P) / p(S)• Posterior = Likelihood * Prior / Evidence• p(P|S)

– Degree of belief in ‘P’, given the data ‘S’ depends on what the data tell you ‘p(S|P’ and any prior information ‘p(P’

100 participants80 non-smokers10 professors4 non-smokers

This year, the conference is held in New York where smokers have to pay extra. This doesn’t deter the professors, but lots of the other participants decide to give up for the week! If you repeat your

indiscretion this year, what are the chances of the smoker at the bar being a professor?

p(P|S) = p(S|P)*p(P) p(S)

= 0.06/0.234 = 0.25

p(P) = 0.1

p(P’) = 0.9

p(S|P) = 0.6

p(S’|P) = 0.4

p(S|P’) = 0.2

p(S’|P’) = 0.8

p(P|S) = p(P and S) p(S)

p(S) = p(S and P) or p(S and P’)

‘AND’ = multiply

‘OR’ = add

p(S and P) = p(S|P)*p(P)

p(S and P’) = p(S|P’)*p(P’)

p(S) = 0.6*0.1 + 0.2*0.9 = 0.234

Bayes in SPM2

• p(yp(y|*p(p(y)

• p(yp(y|*p(• Posterior Probability Map (PPM)

– Posterior Distribution– Likelihood Function (equivalent to normal

SPM)– Prior Probabilities of parameters

• PPM = SPM * priors

Bayes in SPM2

• Deciding on the priors– Fully specified priors in DCM– Estimating priors in PEB

• Computing the Posterior Distribution

• Making inferences– Shrinkage priors– Thresholds

Priors• Everyone has prior beliefs about their data • In Bayesian Framework priors are formally tested

Bayesian Inference

Full Bayes

Previous empirical data eg biophysics of haemodynamic response

Empirical Bayes

Mean & Variance of priors estimated from the data

Hierarchical model: Parameters from one level become the priors at next level

Between voxel variance over all voxels used as prior on variance at each voxel

PPMs: 1st Level = within voxel of interest 2nd Level = between all brain voxels

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

= + z 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 post

Computing the Posterior Probability Distribution

The effects of different precisionsp = d

p < d

p > d

p ≈ 0

Multivariate Distributions

Shrinkage Priors Small, variable effect Large, variable effect

Small, consistent effect Large, consistent effect

Reporting PPMs

• Posterior Distribution describes probability of getting an effect, given the data

• Posterior distribution is different for every voxel• Size of effect (Mean) & Variability (Precision)• 2 Steps

– Decide what size of effect is physiologically relevant– Each voxel 95% certain that the effect size is greater

than threshold

• Special extension of Bayesian Inference for PPMs

Thresholding Small, variable effect Large, variable effect

Small, consistent effect Large, consistent effect

p(| y) = 0.95

Bayesian Inference in SPM2 • Bayesian Inference offers a solution

– All inferences relate to disproving the null hypothesis

• There is no null hypothesis• Different hypotheses can be tested formally

– Multiple Comparisons & False Positives• Voxel wise inferences are independent• P-values don’t change with search volume• Use of shrinkage priors

– Very small effects can be declared significant with enough data

• Thresholding of effect size

With thanks to Klaas & Will