Probability Frequentist versus Bayesian and why it matters Roger Barlow Manchester University 11 th...

ProbabilityFrequentist versus Bayesianand why it mattersRoger Barlow

Manchester University

11th February 2008

Frequentist and Bayesian Probability Roger Barlow Slide 2

Outline Different definitions of probability: Frequentist and

Bayesian Measurements: definitions usually give the same

results Differences in dealing with

Nongaussian measurements Small number counting Constrained parameters

Difficulties for both Conclusions and recommendations


What is Probability?

A is some possible event or fact.

What is P(A)?

Classical: An intrinsic property

Frequentist: Limit N N(A) / N

Bayesian: My degree of belief in A

What do we mean by P(A)?


Classical (Laplace and others)

Symmetry factor Coin – ½ Cards – 1/52

Dice – 1/6

Roulette – 1/32

Equally likely outcomes

Extend to more complicated systems of several coins,

many cards, etc.

The probability of an event is the ratio of the number of cases favourable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders

them, for us, equally possible. Théorie analytique des probabilités


Classical Probability Breaks down

Can’t handle continuous variables

Bertrand’s paradox: if we draw a chord at random, what is the probability that it is longer than the side of the triangle?

Answer: 1/3 or ½ or 1/4

Cannot enumerate ‘equally likely’ cases

in a unique way


Frequentist Probability (von Mises, Fisher) Limit of frequency

P(A)= Limit N N(A)/N

This was a property of the classical definition, now promoted to become a definition itself

P(A) depends not just on A but on the ensemble – which must be specified.

This leads to two surprising features

Ensemble of Everything

A


Feature 1:There can be many Ensembles

Probabilities belong to the event and the ensemble Insurance company data shows P(death) for 40 year old male

clients = 1.4% (example due to von Mises) Does this mean a particular 40 year old German has a 98.6%

chance of reaching his 41st Birthday? No. He belongs to many ensembles

German insured males German males Insured nonsmoking vegetarians German insured male racing drivers …

Each of these gives a different number. All equally valid.


Feature 2: Unique events have no ensemble

Some events are unique.

Consider

“It will probably rain tomorrow.”

There is only one tomorrow (Tuesday 12th February). There is NO ensemble. P(rain) is either 0/1 =0 or 1/1 = 1

Strict frequentists cannot say 'It will probably rain tomorrow'.

This presents severe social problems.


Circumventing the limitationA frequentist can say:

“The statement ‘It will rain tomorrow’ has a 70% probability of being true.”

by assembling an ensemble of statements and ascertaining that 70% (say) are true.

(E.g. Weather forecasts with a verified track record)

Say “It will rain tomorrow” with 70% confidence

For unique events, confidence level statements replace probability statements.


Bayesian (Subjective) Probability

P(A) is a number describing my degree of belief in A1=certain belief. 0=total disbelief Can be calibrated against simple classical

probabilities. P(A)=0.5 means: I would be indifferent given the choice of betting on A or betting on a coin toss.

A can be anything: death, rain, horse races, existence of SUSY…

Very adaptable. But no guarantee my P(A) is the same as your P(A). Subjective = unscientific?


Bayes’ Theorem

“Bayesian” form

P(Theory|Data)=P(Data|Theory) P(Theory)

P(Data)

“Theory” may be an event (e.g. rain tomorrow)

Or a parameter value (e.g. Higgs Mass): then P(Theory) is a function

P(MH|Data)=P(Data|MH) P(MH)

P(Data)PriorPosterior


Measurements:Bayes at work Result value x Theoretical ‘true’ value P(|x) P(x|) P()

Prior is generally taken as uniformIgnore normalisation problems

Construct theory of measurements – prior of second measurement is posterior of the first

P(x|) is often Gaussian, but can be anything (Poisson, etc) For Gaussian measurement and uniform prior, get Gaussian posterior

= X


Aside: “Objective” Bayesian statistics Attempt to lay down rule for choice of prior ‘Uniform’ is not enough. Uniform in what? Suggestion (Jeffreys): uniform in a variable

for which the expected Fisher information <d2ln L/dx2>is minimum (statisticians call this a ‘flat prior’).

Has not met with general agreement – different measurements of the same quantity have different objective priors


Measurement and Frequentist probabilityMT=1743 GeV :What does it mean?

For true value the probability (density) for a result x is (for the usual Gaussian measurement)

P(x ; , )=(1/ 2) exp-[(x -)2/22]For a given , the probability that x lies within is 68%. This

does not mean that for a given x, the ‘inverse’ probability that lies within is 68%

P(x; , ) cannot be used as a probability for .(It is called the likelihood function for given x.)

MT=1743 GeVIs there a 68% probability that MT lies between 171 and 177 GeV?

No. MT is unique. It is either in the range or outside. (Soon we’ll know.)

But 3 does bracket x 68% of the time: The statement ‘MT lies between 171 and 177 GeV’ has a 68% probability of being true.

MT lies between 171 and 177 GeV with 68% confidence


Pause for breath

For Gaussian measurements of quantities with no constraints/objective prior knowledge the same results are given by:

Frequentist confidence intervals Bayesian posteriors from uniform priorsA frequentist and a simple Bayesian will report the

same outcome from the same raw data, except one will say ‘confidence’ and the other ‘probability’. They mean something different but such concerns can be left to the philosophers


Frequentist confidence intervals beyond the simple GaussianSelect Confidence Level value CL and strategyFrom P(x ,) choose construction (functions

x1(), x2()) for which

P(x[x1(), x2()]) CL for all

Given a measurement X, make statement

[LO, HI] @ CL

Where X=x2(LO), X=x1(HI)(Neyman technique)


Confidence Belt

x

Example: proportional Gaussian

= 0.1

Measures with 10% accuracy

Result (say) 100.0

LO=90.91 HI= 111.1

Constructed horizontally such that the probability of a result lying inside the belt is 68%(or whatever)

Read vertically using the measurement

X


Bayesian Proportional Gaussian Likelihood function

C exp(- ½(-100)2/(0.1 )2)

Integration gives C=0.03888

68% (central) limits

92.6 and 113.8

Different techniques give different answers

16%

68%

16%


Small number counting experiments

Poisson distribution P(r; )= e- r / r!

For large can use Gaussian approx. But not small Frequentists: Choose CL. Just use one curve to give

upper limitDiscrete observable makes smooth curves into ugly

staircases

Observe n. Quote upper limit as HI from solving

0n P(r, HI) = 0

n e-HI HI r/r! = 1-CLTranslation. n is small. can’t be very large. If the true

value is HI (or higher) then the chance of a result this small (or smaller) is only (1-CL) (or less)


Frequentist Poisson Table

Upper limitsn 90% 95% 99%0 2.30 3.00 4.611 3.89 4.74 6.642 5.32 6.30 8.413 6.68 7.75 10.054 7.99 9.15 11.605 9.27 10.51 13.11

.....


Bayesian limits from small number countsP(r,)=exp(- ) r/r!

With uniform prior this gives posterior for

Shown for various small r results

Read off intervals...r=6

r=2

r=1

r=0 P()


Upper limitsUpper limit from n events

0

HI exp(- ) n/n! d = CL

Repeated integration by parts:0

n exp(- HI) HIr/r! = 1-CL

Same as frequentist limit This is a coincidence! Lower Limit formula is not

the same


Result depends on Prior

Example: 90% CL Limit from 0 events

Prior flat in

Prior flat in

X

X =

=1.65

2.30


Which is right?

Bayesian Method is generally easier, conceptually and in practice

Frequentist method is truly objective. Bayesian probability is personal ‘degree of belief’. This does not worry biologists but should worry physicists.

Ambiguity appears in Bayesian results as differences in prior give different answers, though with enough data these differences vanish

Check for ‘robustnesss under change of prior’ is standard statistical technique, generally ignored by physicists

‘Uniform priors’ is not a sufficient answer. Uniform in what?


Problems for Frequentists:Add a background =S+bFrequentist method (b known, measured, S wanted 1. Find range for 2. Subtract b to get range for SExamples: See 5 events, background 1.2

95% Upper limit: 10.5 9.3 See 5 events, background 5.1

95% Upper limit: 10.5 5.4 ? See 5 events, background 10.6

95% Upper limit: 10.5 -0.1


S< -0.1? What’s going on?

If N<b we know that there is a downward fluctuation in the background. (Which happens…)

But there is no way of incorporating this information without messing up the ensemble

Really strict frequentist procedure is to go ahead and publish.

We know that 5% of 95%CL statements are wrong – this is one of them

Suppressing this publication will bias the global results


Similar problems Expected number of events must be non-negative Mass of an object must be non-negative Mass-squared of an object must be non-negative Higgs mass from EW fits must be bigger than LEP2

limit of 114 GeV

3 Solutions Publish a ‘clearly crazy’ result Use Feldman-Cousins technique Switch to Bayesian analysis


=S+b for Bayesians No problem! Prior for is uniform for Sb Multiply and normalise as before

Posterior Likelihood Prior

Read off Confidence Levels by integrating posterior

=X


Another Aside: Coverage

Given P(x;) and an ensemble of possible measurements {xi} and some confidence level algorithm, coverage is how often ‘

LO HI’ is true.Isn’t that just the confidence level? Not quite.

Discrete observables may mean the confidence belt is not exact – move on side of caution

Other ‘nuisance’ parameters may need to be taken account of – again erring on side of caution

Coverage depends on . For a frequentist it is never less than the CL (‘undercoverage’). It may be more (‘overcoverage’) – this is to be minimised but not crucial

For a Bayesian coverage is technically irrelevant – but in practice useful


Bayesian pitfall(Heinrich and others)Observe n events from Poisson with =S+bChannel strength S – unknown. Flat priorEfficiency x luminosity - from sub-experiment

with flat priorBackground b - from sub-experiment with flat

priorInvestigated coverage and all OKPartition into classes (e.g. different run periods)Coverage falls!


What’s happening

Problem due to efficiency x Lumi priors

= 1+ 2 + 3 +…Uniform density in all N

components means P () N-1

‘Solve’ by taking priors P(i) 1/ i (Arguments you should have

done so in the first place – Jeffreys’ Prior)

2

1


Another example: Unitarity triangleMeasure CKM angle by measuring B decays

(charged and neutral, branching ratios and CP asymmetries). 6 quantities.

Many different parametrisations suggested

Uniform priors in different parametrisations give different results from each other and from a Frequentist analysis (according to CKMfitter: disputed by UTfit)

For a complex number z=x+iy=rei a flat prior in x and y is not the same as a flat prior in r and


Loss of ambiguities?

Toy exampleMeasure X=(+)2=1.00 0.07, Y= 2=1.100.07 Is interesting but is a nuisance parameterClearly 4 fold ambiguity 1, 0 or 2 Frequentists stop thereBayesians integrate over and get a peak at 0

double that at 2 – feature persists whatever the prior used for .

Is this valid? (Real example will be more subtle)


ConclusionsFrequentist statistics cannot do everythingBayesian statistics can be dangerous. Choose

between1) Never use it2) Use only if frequentist method has problems3) Use only with care and expert guidance and always

check for robustness under different priors4) Use as investigative tool to explore possible

interpretations5) Just plug in the package and write down the resultsBut always know what you are doing and say what you

are doing.


Backup slides


Incorporating Constraints: PoissonWork with total source strength (s+b) you know

is greater than the background b

Need to solve

Formula not as obvious as it looks.

1 CL 0

ne s b s b r r !

0

ne b b r r !


Feldman Cousins MethodWorks by attacking what looks like a different problem...

E xampl e:

Y ou hav e a back grou n d of 3 .2

O bserv e 5 ev en t s? Q u ot e on e-si ded u pper l i mi t (9 .27 -3 .2 =6.07@ 90% )

O bserv e 25 ev en t s? Q u ot e t w o-si ded l i mi t s

Physic ist s ar e humanI deal Physic ist1. Choose S t r at egy2 . E x amine dat a3 . Q uot e r esult

Real Physic ist1. E x amine dat a2 . Choose S t r at egy3 . Q uot e R esult

A ls o c a lle d * ‘th e U n ifie d A p p ro a c h ’

* by Feldman and Cousins, mostly


Feldman Cousins: =s+bb is known. N is measured. s is what we're after

This is called 'flip-flopping' and BAD because is wrecks the whole design of the Confidence Belt

Suggested solution:

1) Construct belts at chosen CL as before

2) Find new ranking strategy to determine what's inside and what's outside

1 sided 90%

2 sided 90%


Feldman Cousins: RankingFirst idea (almost right)Sum/integrate over range of (s+b) values with

highest probabilities for this observed N.(advantage that this is the shortest interval)

Glitch: Suppose N small. (low fluctuation)P(N;s+b) will be small for any s and never get

countedInstead: compare to 'best' probability for this N, at

s=N-b or s=0 and rank on that numberSuch a plot does an automatic ‘flip-flop’N~b single sided limit (upper bound) for sN>>b 2 sided limits for s


How it worksHas to be computed for the

appropriate value of background b. (Sounds complicated, but there is lots of software around)

As n increases, flips from 1-sided to 2-sided limits – but in such a way that the probability of being in the belt is preserved

s

n

Means that sensible 1-sided limits are quoted instead of nonsensical 2-sided limits!


Arguments against using Feldman Cousins

Argument 1

It takes control out of hands of physicist. You might want to quote a 2 sided limit for an expected process, an upper limit for something weird

Counter argument:

This is the virtue of the method. This control invalidates the conventional technique. The physicist can use their discretion over the CL. In rare cases it is permissible to say ”We set a 2 sided limit, but we're not claiming a signal”


Feldman Cousins: Argument 2 Argument 2

If zero events are observed by two experiments, the one with the higher background b will quote the lower limit. This is unfair to hardworking physicists

Counterargument

An experiment with higher background has to be ‘lucky’ to get zero events. Luckier experiments will always quote better limits. Averaging over luck, lower values of b get lower limits to report.

Example: you reward a good student with a lottery ticket which has a 10% chance of winning £10. A moderate student gets a ticket with a 1% chance of winning £ 20. They both win. Were you unfair?


3. Including Systematic Errors =aS+b is predicted number of eventsS is (unknown) signal source strength.

Probably a cross section or branching ratio or decay rate

a is an acceptance/luminosity factor known with some (systematic) error

b is the background rate, known with some (systematic) error


3.1 Full Bayesian

Assume priors for S (uniform?) For a (Gaussian?) For b (Poisson or Gaussian?)

Write down the posterior P(S,a,b).

Integrate over all a,b to get marginalised P(s)

Read off desired limits by integration


3.2 Hybrid Bayesian

Assume priors For a (Gaussian?) For b (Poisson or Gaussian?)Integrate over all a,b to get marginalised P(r,S)

Read off desired limits by 0nP(r,S) =1-CL etc

Done approximately for small errors (Cousins and Highland). Shows that limits pretty insensitive to a , b

Numerically for general errors (RB: java applet on SLAC web page). Includes 3 priors (for a) that give slightly different results


3.3-3.9

Extend Feldman Cousins Profile Likelihood: Use P(S)=P(n,S,amax,bmax)

where amax,bmax give maximum for this S,n Empirical Bayes And more…

Results being compared as outcome from Banff workshop


Summary Straight Frequentist approach is objective and

clean but sometimes gives ‘crazy’ results Bayesian approach is valuable but has

problems. Check for robustness under choice of prior

Feldman-Cousins deserves more widespread adoption

Lots of work still going on This will all be needed at the LHC

Probability Frequentist versus Bayesian and why it matters Roger Barlow Manchester University 11 th...

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