The populist view of probability is the so-called frequentist approach:

whereby the probability P of an uncertain event A, P(A), is defined by the frequency of that event based on previous observations.

For example, in the UK 50.9% of all babies born are girls; suppose then that we are interested in the event A: 'a randomly selected baby is a girl'.

According to the frequentist approach P(A)=0.509.

Bayesianism

The frequentist approach for defining the probability of an uncertain event is fine providing that we have been able to record accurate information about many past instances of the event. However, if no such historical database exists, then we have to consider a different approach.

Bayesian probability is a formalism that allows us to reason about beliefs under conditions of uncertainty. If we have observed that a particular event has happened, such as Britain coming 10th in the medal table at the 2004 Olympics, then there is no uncertainty about it.

However, suppose a is the statement “Britain sweeps the boards at 2012 London Olympics, winning 36 Gold Medals!“

Since this is a statement about a future event, nobody can state with any certainty whether or not it is true. Different people may have different beliefs in the statement depending on their specific knowledge of factors that might effect its likelihood.

Sporting woes continued…

For example, Henry may have a strong belief in the statement a based on his knowledge of the current team and past achievements.

Marcel, on the other hand, may have a much weaker belief in the statement based on some inside knowledge about the status of British sport; for example, he might know that British sportsmen failed in bids to qualify for the Euro 2008 in soccer, win the Rugby world cup and win the Formula 1 world championship – all in one weekend!

Thus, in general, a person's subjective belief in a statement a will depend on some body of knowledge K. We write this as P(a|K). Henry's belief in a is different from Marcel's because they are using different K's. However, even if they were using the same K they might still have different beliefs in a.

The expression P(a|K) thus represents a belief measure. Sometimes, for simplicity, when K remains constant we just write P(a), but you must be aware that this is a simplification.

Bayes Theorem

True Bayesians actually consider conditional probabilities as more basic than joint probabilities . It is easy to define P(A|B) without reference to the joint probability P(A,B). To see this note that we can rearrange the conditional probability formula to get:

P(A|B) P(B) = P(A,B)

by symmetry:

P(B|A) P(A) = P(A,B)

It follows that:

which is the so-called Bayes Rule.

Why are we here?

Fundamental difference in the experimental approach

Normally reject (or fail to reject H0) based on an arbitrarily chosen P value (conventionally <0.05) [In other words we choose our willingness to accept a Type I error]

This tells us nothing about the probability of H1

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

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.

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

ToddBridges 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+)

An Example

Suppose that we are interested in diagnosing cancer in patients who visit a chest clinic:

Let A represent the event "Person has cancer"

Let B represent the event "Person is a smoker"

We know the probability of the prior event P(A)=0.1 on the basis of past data (10% of patients entering the clinic turn out to have cancer). We want to compute the probability of the posterior event P(A|B). It is difficult to find this out directly. However, we are likely to know P(B) by considering the percentage of patients who smoke – suppose P(B)=0.5. We are also likely to know P(B|A) by checking from our record the proportion of smokers among those diagnosed. Suppose P(B|A)=0.8.

We can now use Bayes' rule to compute:

P(A|B) = (0.8 * 0.1)/0.5 = 0.16

Thus, in the light of evidence that the person is a smoker we revise our prior probability from 0.1 to a posterior probability of 0.16. This is a significance increase, but it is still unlikely that the person has cancer.

Another Example…

Suppose that we have two bags each containing black and white balls.

One bag contains three times as many white balls as blacks. The other bag contains three times as many black balls as white.

Suppose we choose one of these bags at random. For this bag we select five balls at random, replacing each ball after it has been selected. The result is that we find 4 white balls and one black.

What is the probability that we were using the bag with mainly white balls?

Solution

Solution. Let A be the random variable "bag chosen" then A={a1,a2} where a1 represents "bag with mostly white balls" and a2 represents "bag with mostly black balls" . We know that P(a1)=P(a2)=1/2 since we choose the bag at random.

Let B be the event "4 white balls and one black ball chosen from 5 selections".

Then we have to calculate P(a1|B). From Bayes' rule this is:

Now, for the bag with mostly white balls the probability of a ball being white is ¾ and the probability of a ball being black is ¼. Thus, we can use the Binomial Theorem, to compute P(B|a1) as:

Similarly

hence

A big advantage of a Bayesian approach

Allows a principled approach to the exploitation of all available data …

with an emphasis on continually updating one’s models as data accumulate

as seen in the consideration of what is learned from a positive mammogram

Bayesian Reasoning

ASSUMPTIONS

1% of women aged forty who participate in a routine screening have breast cancer

80% of women with breast cancer will get positive tests

9.6% of women without breast cancer will also get positive tests

EVIDENCE

A woman in this age group had a positive test in a routine screening

PROBLEM

What’s the probability that she has breast cancer?

Bayesian Reasoning

ASSUMPTIONS

10 out of 1000 women aged forty who participate in a routine screening have breast cancer

800 out of 1000 of women with breast cancer will get positive tests

95 out of 1000 women without breast cancer will also get positive tests

PROBLEM

If 1000 women in this age group undergo a routine screening, about what fraction of women with positive tests will actually have breast cancer?

Bayesian Reasoning

ASSUMPTIONS

100 out of 10,000 women aged forty who participate in a routine screening have breast cancer

80 of every 100 women with breast cancer will get positive tests

950 out of 9,900 women without breast cancer will also get positive tests

PROBLEM

If 10,000 women in this age group undergo a routine screening, about what fraction of women with positive tests will actually have breast cancer?

The conditional probability slides the revised probability in its direction but doesn’t replace the prior probability

A NATURAL FREQUENCIES presentation is one in which the information about the prior probability is embedded in the conditional probabilities (the proportion of people using Bayesian reasoning rises to around half).

Test sensitivity issue (or: “if two conditional probabilities are equal, the revised probability equals the prior probability”)

Where do the priors come from?

-----> Bayes’ theorem

p(X|A)*p(A)

p(A|X) = ______________________

P(X|A)*p(A) + p(X|~A)*p(~A)

Given some phenomenon A that we want to investigate, and an observation X that is evidence about A, we can update the original probability of A, given the new evidence X.

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.

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.”

Chapter from Wolpert DM & Ghahramani Z (In press): Bayes rule in perception, action and cognition Oxford Companion to Consciousness Wolpert DM & Ghahramani Z (In press)

http://bayes.wustl.edu/ Good links for Probability Theory relating to logic

http://www.stat.ucla.edu/history/essay.pdf (Bayes’ original essay!!!)

Bayesian constraints applied: priors based on known ranges of the transformations

Empirically generated priors

Effect of Bayesian constraints on non-linear warping

Affine registration

Non-linear

registration

with

regularisation

(Bayesian constraints)

Template

image

Non-linear

registration

without

regularisation

introduces

unnecessary warps

Surely it’s time for some equations?

Bayes’ rule is used to constrain the non-linear warping by incorporating prior knowledge of the likely extent of deformations:

p(p|e) p(e|p) p(p)

p(p|e) is the posterior probability of parameters p given errors e

p(e|p) is the likelihood of observing errors e given parameters p

p(p) is the prior probability of parameters p

Image Segmentation

Segmentation = dividing up the brain into grey/white matter, cerebro-spinal fluid (CSF), non-brain tissue (skull, etc.)

Used during co-registration (part of pre-processing) and for Voxel Based Morphometry (VBM)

Relies on Bayes’ theorem

Priors are used to constrain image segmentation

Prior probability images taken from average of 152 brains used as Bayesian constraints on image segmentation

Priors:

Image:

Non-brain/skull

CSF

WM

GM

Analysis and Inference: PPMs vs. SPMs

Classical vs. Bayesian inference

Classical: p(these data|H0)

H0 = no experimental effect, i.e. β = 0

Remember that β (beta) is the parameter that indicates the contribution of a particular column in your design matrix to the data:

=

+

x1

β1

β2

β3

ε

y

x2

x3

Observed data = Predictors * Parameters + Error

Cast your mind back to last week…

The betas indicate the contribution of each predictor to the data in each voxel

≈

β1∙

+ β2∙

+ β3∙

2

3

4

0

1

0

1

0

1

2

0.83

0.16

2.98

Listening

Reading

Rest

Classical vs. Bayesian inference

So in Classical inference we ask:

p(these data|β=0) or p(y|β)

In Bayesian inference we ask:

p(β|these data) or p(β|y)

[Remember: p(y|β) ≠ p(β|y)]

p(β|y) p(y|β)*p(β)

posterior likelihood * prior

PPM SPM * priors

This produces the values for a statistical parametric map (SPM)

This is plotted as a posterior probability map (PPM)

So what are the priors?

The SPM program uses Parametric Empirical Bayes (PEB)

Empirical = the priors are estimated from the data

Hierarchical: higher levels of analysis provide Bayesian constraints on lower levels

Typically 3 levels:

within-voxel – between-voxels – between-subjects

constrains

constrains

PEB = classical inference at the highest level

What constrains your highest level?

Parameters unknown so priors are flat

So PEB at the highest level = classical approach

BUT important differences between an SPM and a PPM:

An SPM has uniform specificity (specificity = 1-α)

A PPM has uniform effect size with uniform confidence because it varies voxel-by-voxel with the variance of the priors

Putting it another way: PPMs vs. SPMs

When you threshold a PPM you are specifying a desired effect size

But for an SPM, if a voxel survives thresholding it could be a big effect with relatively high variance but it could also be a small effect with low variance…

… and as you increase the number of scans and/or subjects, probability of a very small effect surviving increases

PPMs vs. SPMs continued

Bayesian inference is generally more specific than classical inference (except when variance of priors is very large)

The posterior probability is the same irrespective of whether one voxel or the entire brain was analysed…

…so no multiple comparisons problem!

Wonderful! Why doesn’t everyone use it?

Disadvantages:

Computationally demanding

Not yet readily accepted technique in the neuroimaging community?

It’s not magic. It isn’t better than classical inference for a single voxel or subject, but it is the best estimate on average over voxels or subjects

Bayesian Estimation in SPM5

Other Bayesian applications in neuroimaging

“Variational Bayes” – new to SPM5, constrains data at the voxel (1st) level using “shrinkage priors” (assume overall effect is zero) with prior precision estimated from the data for each brain slice

Dynamic Causal Modelling (DCM) uses Bayesian constraints on the connections between brain areas and their dynamics

EEG and MEG use Bayesian constraints on source reconstruction