The great Bayesian vs. Frequentist war has raged within statistics for almost 100 years, much to the confusion of outsiders. The Bayesian/Frequentist question is no longer academic, with both styles of inference appearing frequently in scientific literature and even the news. In this talk, Kristin Lennox aims to explain the great divide to non-statisticians, and also to answer the most important statistical question of all: how does probability allow us to better understand our world?

1. LLNL-PRES-697098
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore
National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC
All
About
That
Bayes
Probability,
Sta4s4cs,
and
the
Quest
to
Quan4fy
Uncertainty
Kris%n
P.
Lennox
Director
of
Sta%s%cal
Consul%ng
July 28, 2016

2. 2
LLNL-PRES-697098
Man
of
the
(Literal)
Hour
Probably not Thomas Bayes, but often mistaken for him
Source: Wikipedia

3. 3
LLNL-PRES-697098
Central
Dogma
of
Inferen4al
Sta4s4cs
Statisticians use probability to describe
uncertainty.

4. 4
LLNL-PRES-697098
You
Are
Here
Why This Matters
What is Probability?
What is Uncertainty?
An Incomplete History of Uncertainty Quantification
The BIG Reveal

5. 5
LLNL-PRES-697098
You
Are
Here
Why This Matters
What is Probability?
What is Uncertainty?
An Incomplete History of Uncertainty Quantification
The BIG Reveal

6. 6
LLNL-PRES-697098
§ Probability
§ Distribu%on
§ Parameter
§ Likelihood
What
is
Probability?
1933
A. N. Kolmogorov
Copyright MFO, Creative Commons License

7. 7
LLNL-PRES-697098
A. N. Kolmogorov
Copyright MFO, Creative Commons License
§ Probability
is
a
measure.
§ Distribu%on
§ Parameter
§ Likelihood
What
is
Probability?
1933

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LLNL-PRES-697098
§ Probability
is
a
measure.
§ Distribu%ons
define
measure
of
events.
§ Parameter
§ Likelihood
Exponential Normal/Gaussian
What
is
Probability?

9. 9
LLNL-PRES-697098
§ Probability
is
a
measure.
§ Distribu%ons
define
measure
of
events.
§ Parameters
define
distribu%ons.
§ Likelihood
Exponential Normal/Gaussian
What
is
Probability?

10. 10
LLNL-PRES-697098
f (x) = Pr(X = x |Θ =θ)
§ Probability
is
a
measure.
§ Distribu%ons
define
measure
of
events.
§ Parameters
define
distribu%ons.
§ Likelihood
fixes
data
and
varies
parameters.
What
is
Probability?

11. 11
LLNL-PRES-697098
§ Probability
is
a
measure.
§ Distribu%ons
define
measure
of
events.
§ Parameters
define
distribu%ons.
§ Likelihood
fixes
data
and
varies
parameters.
l(θ) = Pr(X = x |Θ =θ)
What
is
Probability?

12. 12
LLNL-PRES-697098
You
Are
Here
Why This Matters
What is Probability?
What is Uncertainty?
An Incomplete History of Uncertainty Quantification
The BIG Reveal

13. 13
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15
1

14. 14
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15
2

15. 15
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15

16. 16
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15

17. 17
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15

18. 18
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15
3

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LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15

20. 20
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15

21. 21
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15

22. 22
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15
4

23. 23
LLNL-PRES-697098
A
Fable
The
Sta4s4cal
Lunch
Bunch
and
the
Summer
Student
Revolt
of
‘15

24. 24
LLNL-PRES-697098
You
Are
Here
Why This Matters
What is Probability?
What is Uncertainty?
An Incomplete History of Uncertainty Quantification
The BIG Reveal

25. 25
LLNL-PRES-697098
In
the
Beginning…
1654
P. Fermat
Source: Wikipedia, Creative Commons License
B. Pascal
Source: Wikipedia, Creative Commons License

26. 26
LLNL-PRES-697098
Thomas
Bayes
and
the
Doctrine
of
Chances
1763
Still not Thomas Bayes
Source: Wikipedia

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Blindfolded
1-­‐Dimensional
Table
Bocce

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LLNL-PRES-697098
Blindfolded
1-­‐Dimensional
Table
Bocce
PriorDensity
n=0

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LLNL-PRES-697098
Blindfolded
1-­‐Dimensional
Table
Bocce
PosteriorDensity
n=1

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LLNL-PRES-697098
PosteriorDensity
Blindfolded
1-­‐Dimensional
Table
Bocce
n=2

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LLNL-PRES-697098
PosteriorDensity
Blindfolded
1-­‐Dimensional
Table
Bocce
n=10

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LLNL-PRES-697098
PosteriorDensity
Blindfolded
1-­‐Dimensional
Table
Bocce
n=25

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LLNL-PRES-697098
§ What
is
the
probability
that
event
x
occurs
given
that
event
y
occurs?
Bayes
Theorem
Pr(X = x |Y = y) =
Pr(Y = y | X = x)Pr(X = x)
Pr(Y = y)

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LLNL-PRES-697098
What
is
the
probability
that
our
distribu,on
parameter
is
θ given
that
we
have
observed
data
x?
Bayes
Theorem
–
Bayesian
Version
Pr(Θ =θ | X = x) =
Pr(X = x |Θ =θ)Pr(Θ =θ)
Pr(X = x)
Prior distribution of θ
Posterior distribution of θ given x
A constant of integration that most people don’t talk about
Likelihood

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LLNL-PRES-697098
The
Man
Who
Invented
Sta4s4cs
18th
Century
P. S. Laplace
Source: Wikipedia, Creative Commons License

36. 36
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The
Prior
and
Its
Enemies
Pr(Θ =θ | X = x)∝ Pr(X = x |Θ =θ)Pr(Θ =θ)
Posterior distribution of θ given x Likelihood
Where does the prior come from?
Prior distribution of θ

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LLNL-PRES-697098
Pr(Θ =θ | X = x)∝ Pr(X = x |Θ =θ)Pr(Θ =θ)
Prior distribution of θPosterior distribution of θ given x Likelihood
What is the probability that the sun will rise
tomorrow?
The
Sun
Will
Come
Out
Tomorrow?

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LLNL-PRES-697098
Pr(Θ =θ | X = x)∝ Pr(X = x |Θ =θ)
Posterior distribution of θ given x Likelihood
What is the probability that the sun will rise
tomorrow?
The
Sun
Will
Come
Out
Tomorrow?
in (0,1)

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LLNL-PRES-697098
The
Sun
Will
Come
Out
Tomorrow?
What is the probability that the sun will rise
tomorrow?
E θ | X = x[ ]=
# of times sun has come up + 1
# of times sun has come up + 2

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LLNL-PRES-697098
The
Sun
Will
Come
Out
Tomorrow?*
What is the probability that the sun will rise
tomorrow?
= 0.9999995=
5000×365.25 + 1
5000×365.25 + 2
*As calculated by Laplace

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LLNL-PRES-697098
The
Frequen4sts
Early
20th
Century
J. Neyman
Source: Wikipedia, Creative Commons License
R. A. Fisher
Source: Wikipedia, Creative Commons License

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Case
Study:
Interval
Es4ma4on
What is lowest reasonable Pr(X=1)?

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LLNL-PRES-697098
Case
Study:
Interval
Es4ma4on
Bayesian
Solu%on
(Credible
Interval)
1. Pick
a
prior.

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LLNL-PRES-697098
Case
Study:
Interval
Es4ma4on
Bayesian
Solu%on
(Credible
Interval)
1. Pick
a
prior.
2. Calculate
posterior.

45. 45
LLNL-PRES-697098
Bayesian
Solu%on
(Credible
Interval)
1. Pick
a
prior.
2. Calculate
posterior.
3. Find
5th
percen%le.
Case
Study:
Interval
Es4ma4on
95%
(subjec%ve)
probability
that
Pr(X=1)
is
at
least
15.3%.

46. 46
LLNL-PRES-697098
Case
Study:
Interval
Es4ma4on
Frequen%st
Solu%on
(Confidence
Interval)
1. Determine
all
results
that
are
as
or
more
consistent
with
outcome
of
interest.
Care
about
12
or
more
1s
in
50
rolls.

47. 47
LLNL-PRES-697098
Frequen%st
Solu%on
(Confidence
Interval)
1. Determine
all
results
that
are
as
or
more
consistent
with
outcome
of
interest.
2. Iden%fy
all
Pr(X=1)
that
have
>5%
chance
of
producing
12
or
more
1s.
With
95%
confidence,
Pr(X=1)
is
at
least
14.5%.
Case
Study:
Interval
Es4ma4on
Pr(12ormore1sin50rolls)
Pr(X=1)

48. 48
LLNL-PRES-697098
Case
Study:
Interval
Es4ma4on
Both
give
Pr(X=1)
>15%*
with
“uncertainty”
of
5%.
*15.3%
(Bayesian
with
Jeffreys
prior)
vs.
14.5%
(frequen%st)
…
but
confidence
interval
takes
a
lot
longer
to
explain.

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LLNL-PRES-697098
Why
Did
This
Catch
On?
Objec%ve
probability
is
restric%ve,
but
results
mean
the
same
thing
to
everyone.
(Even
if
you
don’t
know
what
they
mean.)

50. 50
LLNL-PRES-697098
BaXle
of
the
Bayesians
20th
Century
-­‐
???
vs
Representative likeness of a subjective BayesianRepresentative likeness of an objective Bayesian

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The
Search
For
Scorpion
1968

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LLNL-PRES-697098
The
Search
For
Scorpion
1968
Contact M8/3

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LLNL-PRES-697098
The
Search
For
Scorpion
1968
Scorpion location

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LLNL-PRES-697098
Computa4on
Late
20th
Century

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LLNL-PRES-697098
You
Are
Here
Why This Matters
What is Probability?
What is Uncertainty?
An Incomplete History of Uncertainty Quantification
The BIG Reveal

56. 56
LLNL-PRES-697098
My
Uncertainty
Quan4fica4on
Toolbox

57. 57
LLNL-PRES-697098
What
Have
We
Learned
Today?
Statisticians use probability to describe
uncertainty.
We do not always agree about
how this should be done.

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LLNL-PRES-697098
Ques%ons?
lennox@llnl.gov
Thank
you
for
coming!
Thomas Bayes would never have worn these glasses.

59. 59
LLNL-PRES-697098
§ The
Theory
That
Would
Not
Die:
How
Bayes'
Rule
Cracked
the
Enigma
Code,
Hunted
Down
Russian
Submarines,
and
Emerged
Triumphant
from
Two
Centuries
of
Controversy.
McGrayne,
S.
B.
Yale
University
Press.
(2011)
§ Bruno
de
FineJ:
Radical
Probabilist.
Ed.
Galavoj,
M.
C.
Texts
in
Philosophy
8.
College
Publica%ons.
(2009)
§ Fisher,
Neyman,
and
the
Crea,on
of
Classical
Sta,s,cs.
Lehmann,
E.
L.
Springer.
(2011)
§ The
History
of
Probability
and
Sta,s,cs
and
Their
Applica,ons
Before
1750.
Hald,
A.
Wiley-­‐Interscience.
(2003)
§ Founda,ons
of
the
Theory
of
Probability.
Kolmogorov,
A.
N.
2nd
English
Edi%on.
Chelsea
Publishing
Company.
(1950)
§ “Opera%ons
Analysis
During
the
Underwater
Search
for
Scorpion.”
Richardson,
H.
R.
and
Stone,
L.
D.
Naval
Research
Quarterly.
18,
pp.
141-­‐157
(1971)
§ “An
Essay
Towards
Solving
a
Problem
in
the
Doctrine
of
Chances.”
Bayes,
T.
and
Price,
R.
Philosophical
Transac,ons.
53,
pp.
370-­‐418
(1763)
§ “Sta%s%cal
Analysis
and
the
Illusion
of
Objec%vity.”
Berger,
J.
and
Berry,
D.
American
Scien,st.
76,
pp.
159-­‐165
(1988)
§ “You
May
Believe
You
are
a
Bayesian
but
You
are
Probably
Wrong.”
Senn,
S.
Ra,onality,
Markets
and
Morals.
2,
pp.
48-­‐66
(2011)
§ “The
Case
for
Objec%ve
Bayes.”
Berger,
J.
Bayesian
Analysis.
1,
pp.
1-­‐17
(2004)
§ “When
Genius
Errs:
R.A.
Fisher
and
the
Lung
Cancer
Controversy.”
Stoley,
P.
D.
American
Journal
of
Epidemiology.
133,
pp.
416-­‐425
(1991)
§ “The
Evolu%on
of
Markov
Chain
Monte
Carlo
Methods.”
Richey,
M.
The
American
Mathema,cal
Monthly.
117,
383-­‐413
(May
2010)
§ and
of
course
Wikipedia.org
Further
Reading