1. What is Probability? Bayes and Frequentism
Louis Lyons
Imperial College and Oxford
(CMS experiment at CERN’s LHC)
Islamabad
Aug 2017

2. Topics Will discuss mainly in context of PARAMETER ESTIMATION.
Misconceptions about Probability
Different views of Probability
Bayes versus Frequentism
Conditional Probability and Bayes Theorem
Bayesian priors
P(A;B)  P(B;A)
Examples: Peasant and Dog
Frequentist approach
Meaning of confidence intervals for parameters
Comparison summary of Bayes and Frequentist approaches
Conclusion
Will discuss mainly in context of PARAMETER ESTIMATION.
Also important for GOODNESS of FIT and HYPOTHESIS TESTING

3. Misconceptions
Conversation with my barber about lottery: LL: Would you choose numbers 1,2,3,4,5,6? Barber: No that’s very unlikely to win LL: Lot’s of people think that, but any specific sequence is equally likely, so choose that to avoid having to share the prize. FACT: The set of numbers 1,2,3,4,5,6 is the most popular, so even if that wins, you don’t get a large sum CONCLUSION: My barber was right! Don’t choose 1,2,3,4,5,6

4. Misconceptions
What is probability that LL will meet Director of National Centre for Physics somewhere in 2018? 3% What is probability that LL will meet Director of National Centre for Physics here in 2018? 10%

5. Misconceptions
What is probability that LL will meet Director of National Centre for Physics somewhere in 2018? 3% What is probability that LL will meet Director of National Centre for Physics here in 2018? 10% These are INCONSISTENT

6. PROBABILITY MATHEMATICAL Formal Based on Axioms FREQUENTIST
Ratio of frequencies as n infinity
Repeated “identical” trials
Not applicable to single event or physical constant
BAYESIAN Degree of belief
Can be applied to single event or physical constant
(even though these have unique truth)
Varies from person to person ***
Quantified by “fair bet”
LEGAL “We have heard the expert Statistician, but he told us only the
mathematical probability, while we want the true probability.”

7. Example of ‘analysing data’
Measure height of everyone in this room
Is data consistent with bell-shaped curve (Gaussian or normal distribution)?
Goodness of Fit (2 , etc.)
What are the best values for the position and the width
of the bell-shaped curve? (e.g. 1.71m and 0.16m)
Parameter Determination (Likelihood, 2, Bayes, Frequentist….}
Does data fit better to one Gaussian , or to 2 Gaussians?
Hypothesis Testing (p-values, L-ratios, Bayes,…)

8. Example of ‘analysing data’
Measure height of everyone in this room
Is data consistent with bell-shaped curve (Gaussian or normal distribution)?
What are the best values for the position and the width
of the bell-shaped curve? (e.g. 1.71m and 0.16m)
c) Does data fit better to one Gaussian , or to 2 Gaussians?

9. BAYES and FREQUENTISM: The Return of an Old Controversy
Louis Lyons
Imperial College and Oxford University

11. It is possible to spend a lifetime analysing data without realising that there are two very different fundamental approaches to statistics:
Bayesianism and Frequentism.

12. How can textbooks not even mention Bayes / Frequentism?
For simplest case
with no constraint on
then
at some probability, for both Bayes and Frequentist (but different interpretations)
BUT in other situations (non-Gaussian, excluded ranges for params,
allow for systematics,…..), B and F can and do differ.
See Bob Cousins “Why isn’t every physicist a Bayesian?” Amer Jrnl Phys 63(1995)398

13. We need to make a statement about
Parameters, Given Data
The basic difference between the two:
Bayesian : Probability (parameter, given data)
(an anathema to a Frequentist!)
Frequentist : Probability (data, given parameter)
(a likelihood function)

14. CONDITIONAL PROBABILITY P(A|B) = Prob of A, given that B has occurred
P[A+B] = N(A+B)/Ntot
= {N(A+B)/N(B)} x {N(B)/Ntot}
= P(A|B) x P(B)
If A and B are independent, P(A|B) = P(A) Venn diagram
 P[A+B] = P(A) x P(B)
e.g. P[rainy + Sunday] = P(rainy) x 1/ INDEP
BUT:
P[rainy + December] ≠ P(rainy) x 1/ INDEP
P[Ee large + Eν large] ≠ P(Ee large) x P(Eν large) INDEP
P[A+B] = P(A|B) x P(B) = P(B|A) x P(A)
 P(A|B) = P(B|A) x P(A) / P(B) ***** Bayes’ Theorem
N.B Usually P(A|B) ≠ P(B|A) Examples later
Ntot
N(A)
. N(B)

15. Bayesian versus Classical
P(A and B) = P(A;B) x P(B) = P(B;A) x P(A)
{If A and B are independent, P(A;B) =P(A),
so P(A and B) = P(A)*P(B) }
e.g. A = event contains t quark
B = event contains W boson
or A = I am in Islamabad
B = I am giving a lecture
P(A;B) = P(B;A) x P(A) /P(B)
Completely uncontroversial, provided….

16. Bayesian p(param | data) α p(data | param) * p(param)
Bayes’ Theorem
p(param | data) α p(data | param) * p(param)
posterior
likelihood
prior
Problems: p(param) Has particular value
“Degree of belief”
Prior What functional form?
Coverage

17. p(parameter) Has specific value
“Degree of Belief”
Credible interval
Prior: What functional form?
Uninformative prior: flat?
In which variable? e.g. m, m2, ln m,….?
Even more problematic with more params
Unimportant if “data overshadows prior”
Important for limits
Subjective or Objective prior?

18. Data overshadows prior
Mass of Z boson (from LEP)
Data overshadows prior

19. Even more important for UPPER LIMITS
Prior
Even more important for UPPER LIMITS

20. Mass-squared of neutrino
Prior = zero in unphysical region

21. Bayesian posterior  intervals
Upper limit Lower limit
Central interval Shortest
Posterior prob
Param
value

22. Upper Limits from Poisson data Expect b = 3.0, observe n events
Ilya Narsky, FNAL CLW 2000
Upper Limits from Poisson data
Expect b = 3.0, observe n events
Upper Limits important for excluding models

23. P (Data;Theory) P (Theory;Data) HIGGS SEARCH at CERN
Is data consistent with Standard Model?
or with Standard Model + Higgs?
End of Sept 2000: Data not very consistent with S.M. Prob (Data ; S.M.) < 1% valid frequentist statement
Turned by the press into: Prob (S.M. ; Data) < 1% and therefore Prob (Higgs ; Data) > 99%
i.e. “It is almost certain that the Higgs has been seen”
THIS IS WRONG!

24. P (Data;Theory) P (Theory;Data)

25. P (Data;Theory) P (Theory;Data)
Theory = male or female
Data = pregnant or not pregnant
P (pregnant ; female) ~ 3%

26. P (Data;Theory) P (Theory;Data)
Theory = male or female
Data = pregnant or not pregnant
P (pregnant ; female) ~ 3%
but
P (female ; pregnant) >>>3%

27. Example 1 : Is coin fair ?
Toss coin: 5 consecutive tails
What is P(unbiased; data) ? i.e. p = ½
Depends on Prior(p)
If village priest: prior ~ δ(p = 1/2)
If stranger in pub: prior ~ 1 for 0 < p <1
(also needs cost function)

28. Example 2 : Particle Identification Try to separate π’s and protons
probability (p tag; real p) = 0.95
probability (π tag; real p) = 0.05
probability (p tag; real π) = 0.10
probability (π tag; real π) = 0.90
Particle gives proton tag. What is it?
Depends on prior = fraction of protons
If proton beam, very likely
If general secondary particles, more even
If pure π beam, ~ 0

29. Peasant and Dog
Dog d has 50% probability of being 100 m. of Peasant p
Peasant p has 50% probability of being within 100m of Dog d ? d p x
River x =0
River x =1 km

30. Given that: a) Dog d has 50% probability of being 100 m. of Peasant,
is it true that: b) Peasant p has 50% probability of being within 100m of Dog d ?
Additional information
Rivers at zero & 1 km. Peasant cannot cross them.
Dog can swim across river - Statement a) still true
If dog at –101 m, Peasant cannot be within 100m of dog
Statement b) untrue

31. Neyman “confidence interval” avoids pdf for
Classical Approach
Neyman “confidence interval” avoids pdf for
Uses only P( x; )
Confidence interval :
P( contains ) =
True for any
Varying intervals from ensemble of experiments
fixed
Gives range of for which observed value was “likely” ( )
Contrast Bayes : Degree of belief = is in

32. Classical (Neyman) Confidence Intervals
Uses only P(data|theory)
μ≥ No prior for μ

33. 90% Classical interval for Gaussian
σ = μ ≥ 0
e.g. m2(νe), length of small object
Other methods have different behaviour at negative x

34. at 90% confidence Frequentist Bayesian and known, but random
unknown, but fixed
Probability statement about and
Frequentist
and known, and fixed
unknown, and random
Probability/credible statement about
Bayesian

35. Frequentism: Specific example
Particle decays exponentially: dn/dt = (1/τ) exp(-t/τ)
Observe 1 decay at time t1: L(τ) = (1/τ) exp(-t1/τ)
Construct 68% central interval
t = .17τ
dn/dt τ t
t = 1.8τ
t t
68% conf. int. for τ from
t1 /1.8  t1 /0.17

36. Bayesian versus Frequentism
Bayesian Frequentist
Basis of method
Bayes Theorem 
Posterior probability distribution
Uses pdf for data,
for fixed parameters
Meaning of probability
Degree of belief
Frequentist definition
Prob of parameters?
Yes
Anathema
Needs prior? No
Choice of interval?
Yes (except F+C)
Data considered
Only data you have
….+ other possible data
Likelihood principle?

37. Bayesian versus Frequentism
Bayesian Frequentist
Ensemble of experiment No
Yes (but often not explicit)
Final statement
Posterior probability distribution
Parameter values 
Data is likely
Unphysical/
empty ranges
Excluded by prior
Can occur
Systematics
Integrate over prior
Extend dimensionality of frequentist construction
Coverage
Unimportant
Built-in
Decision making
Yes (uses cost function)
Not useful

38. Bayesianism versus Frequentism
“Bayesians address the question everyone is interested in, by using assumptions no-one believes”
“Frequentists use impeccable logic to deal with an issue of no interest to anyone”

39. Approach used at LHC
Recommended to use both Frequentist and Bayesian approaches
If agree, that’s good
If disagree, see whether it is just because of different
approaches

40. Final Probability Story
(From David Hand ‘The Improbability Principle: Why coincidences, miracles and rare events happen all the time’)
Someone enters 2 lotteries each week, choosing different sets of numbers for each
One week she has the two sets of winning numbers
BUT she has them the wrong way round
Probability of a ticket winning one lottery ~10-7
Probability of 2 tickets winning 2 lotteries in same week ~ 10-14
But after considering:
enormous number of lottery tickets bought
number of weeks lottery operates
number of lotteries around world
probability of someone having 2 winning tickets somewhere in world during last 30 years is very much larger

41. Particle Physics Resources (Astrophysics/Cosmology has its own too)
Books by Particle Physicists
Barlow, Benkhe, Cowan, James, Lista, Lyons, Roe,…..
PDG: Sections on Probability, Statistics and Monte Carlo simulation.
PHYSTAT meetings
CERN and FNAL 2000 for U.L.
PhyStat-nu, Japan and FNAL 2016
PhyStat-DM in 2018?
Statistics Committees
Collider expts: BaBar, CDF, ATLAS, CMS
RooStats
e.g. Lyons + Moneta at CERN (2016) and at IPMU (2017)
“Too easy to use”

42. Conclusions Do your homework: Try to achieve consensus Good luck.
Before re-inventing the wheel, try to see if Statisticians have already found a solution to your statistics analysis problem.
Don’t use your own square wheel if a circular one already exists.
Try to achieve consensus
Good luck.