First and incomplete draft of slides of a talk on the 2nd edition of Judea Pearl's "Causality"

1. Pearl, Causation, 2
(2009)
Richard D. Gill
Monday 13 May, 2019
Leiden Causality Reading Group

2. Prolegomenon
• I own several hard copies of Pearl Causation 1 (2000), have
never read them
• I first met Judea Pearl in … (a long time ago, at an incredible
workshop in Santa Fé)
• I have been most influenced, regarding causality, by
Jamie Robins. “Counterfactuals” are the only way to go
• I’m interested in causality because … I’m a scientist and a
human being. Interested in having fun, love, nature, justice,
truth, morality, where we came from, where we are going,
life-the-universe-and-everything, …, 42 …

3. First impressions of book
Equations (8.2.1)
tions of exogeneity, but it can (in certain ci
instruments.
By insisting that each upper bound in (8.
bound in (8.14a), we obtain the following tes
P(y1, x1 Ϳ z0) ϩ P(y0, x1 Ϳ z1) Յ 1.
P(y1, x0 Ϳ z0) ϩ P(y0, x0 Ϳ z1) Յ 1,
P(y0, x1 Ϳ z0) ϩ P(y1, x1 Ϳ z1) Յ 1,
P(y0, x0 Ϳ z0) ϩ P(y1, x0 Ϳ z1) Յ 1,

4. Equations (8.21)
• Four linear inequalities in 8 “conditional probabilities”
• Probabilities are normalised and non-negative
• These equations define four faces of a polytope
• Where have you seen something like this before?
• What do we know about polytopes?

6. Figure 5: S = 6, average inner area = 14.41%, average outer area = 10.49%, average corner area = 7.55%
Parliamentary elections:
1000 elections in which 8 parties participated and 3 parties got seats.
Very large electorate voting completely at random.
6 seat parliament, Dutch electoral law.

7. Polytopes
I will mean: bounded convex polytopes
• Intersection of finite number of closed Euclidean half-spaces,
and bounded
• Convex hull of finite number of vertices
• Polytopes have faces, facets, simplicial decomposition, …
• Linear programming, George Dantzig’s simplex algorithm,…

8. Example 2:
2x2x2 Bell experiment
• 2 parties (Alice and Bob)
• Each do a measurement (the spin of a particle)
• Each has 2 possible settings (the spin in a particular
direction)
• Each has 2 possible outcomes (“up”, “down”)

9. X Y
HS T
(b) Causal structure of a famous experi-
ment used by quantum physicists to falsify
assumptions of classical physics; see Sec-
tion 9.5.2.
nt structures that entail inequality constraints.
ome of its variables induces a large set of
d Tian, 2006, Evans, 2012, and references
Alice Bob

10. Outcomes
Bob 1 | Bob 2
Settings
Alice2|Alice1
Alice Setting 1
Bob Setting 2
n(12) trials
Up Down
Up p(uu|12) p(ud|12)
Down p(du|12) p(dd|12)
Alice Setting 1
Bob Setting 1
n(11) trials
Up Down
Up p(uu|11) p(ud|11)
Down p(du|11) p(dd|11)
Alice Setting 2
Bob Setting 2
n(22) trials
Up Down
Up p(uu|22) p(ud|22)
Down p(du|22) p(dd|22)
Alice Setting 2
Bob Setting 1
n(21) trials
Up Down
Up p(uu|21) p(ud|21)
Down p(du|21) p(dd|21)

11. The constraints
• Positivity: p(xy|ab) >= 0 for all x, y, a, b
• Normalisation: sum_x sum_y p(xy|ab) = 1 for all a, b
• No-signalling:
• sum_y p(xy|ab) independent of b for all a
• sum_x p(xy|ab) independent of a for all b
No signalling = No action at a distance, in the “outside world”

12. Example 1:
Randomised clinical trial
with partial non-compliance and an
“instrumental variable” Z (i.c., randomisation)
involving the entire population because it is instrum
262 Imperfect Experiments:
Figure 8.1
dencies in a r
pliance. Z se
mperfect Experiments: Bounding Effects and Counterfactuals
Figure 8.1 Graphical representation of causal depen-
dencies in a randomized clinical trial with partial com-
pliance. Z serves as Instrumental Variable.

13. By insisting that each upper bound in (8.
bound in (8.14a), we obtain the following tes
If any of these inequalities is violated, the in
assumptions underlying our model is viola
P(y1, x1 Ϳ z0) ϩ P(y0, x1 Ϳ z1) Յ 1.
P(y1, x0 Ϳ z0) ϩ P(y0, x0 Ϳ z1) Յ 1,
P(y0, x1 Ϳ z0) ϩ P(y1, x1 Ϳ z1) Յ 1,
P(y0, x0 Ϳ z0) ϩ P(y1, x0 Ϳ z1) Յ 1,

14. Please help me from here…
• …

15. The following slides are from an earlier
attempt at organising the material
• Just to show you a couple of pictures and to start some
further discussion …

16. The following slides are
from an earlier attempt at
organising the material
Just to show you a couple of
pictures and to start some further
discussion …

17. I focus on Pearl 2nd edition Chapter 8 where
(surprise surprise) Bell’s *inequality* turns up.
My target is Section 8.4, formula (8.22) “The Instrumental
Inequality” and the discussion middle of page 275.
My aim was for you to understand Judea Pearl’s writing
better, so we can more easily browse the rest of the book;
not to convert you to quantum statistics.
If you *do* want to better understand the quantum
information application of the Instrumental
Inequality - Bell’s *theorem* - I refer you to my one and
only Statistical Science paper (2014)
“Statistics, Causality and Bell's Theorem”
https://arxiv.org/abs/1207.5103

20. Development of Western science is based on two
great achievements: the invention of the formal
logical system (in Euclidean geometry) by the
Greek philosophers, and the discovery of the
possibility to find out causal relationships by
systematic experiment (during the Renaissance).
Albert Einstein (1953)
TO DANNY
AND THE GLOWING AUDACITY OF GOODNESS
Biology (Chakraborty 2001), The Philosophical Review (Hitchcock 2001), Intelligence
(O’Rourke 2001), Journal of Marketing Research (Rigdon 2002), Tijdschrlft Voor (Decock
2002), Psychometrika (McDonald 2002b), International Statistical Review (Lindley 2002),
Journal of Economic Methodology (Leroy 2002), Statistics in Medicine (Didelez 2002),
Journal of Economic Literature (Swanson 2002), Journal of Mathematical Psychology
(Butler 2002), IIE Transactions (Gursoy 2002), Royal Economic Society (Hoover 2003),
Econometric Theory (Neuberg 2003), Economica (Abbring 2003), Economics and
Philosophy (Woodward 2003), Sociological Methods and Research (Morgan 2004),
Review of Social Economy (Boumans 2004), Journal of the American Statistical
Association (Hadlock 2005), and Artificial Intelligence (Kyburg 2005).
Thanks also go to the contributors to UCLA’s Causality blog (http://www.mii.
ucla.edu/causality/) and to William Hsu, the blog’s curator.
A special word of appreciation and admiration goes to Dennis Lindley, who went to
the trouble of studying my ideas from first principles, and allowed me to conclude that
readers from the statistics-based sciences would benefit from this book. I am fortunate
that our paths have crossed and that I was able to witness the intellect, curiosity, and
integrity of a true gentleman.
This chapter could not have been written without the support and insight of Jin Tian,
Avin Chen, Carlo Brito, Blai Bonet, Mark Hopkins, Ilya Shpitser, Azaria Paz, Manabu
Kuroki, Zhihong Cai, Kaoru Mulvihill, and all members of the Cognitive System
Laboratory at UCLA who, in the past six years, continued to explore the green pastures
of causation while I was summoned to mend a world that took the life of my son Daniel
(murdered by extremists in Karachi, Pakistan, 2002). These years have convinced me,
beyond a shadow of doubt, that the forces of reason and enlightenment will win over
fanaticism and inhumanity.
Finally, I dedicate this edition to my wife, Ruth, for her strength, love, and guidance
throughout our ordeal, to my daughters, Tamara and Michelle, and my grandchildren,