The Flow of Association and Causation in Graphs

The Flow of Association and Causation in Graphs

Brady Neal

causalcourse.com

http://causalcourse.com/

Brady Neal / 352

Graph terminology

Bayesian networks and causal graphs

The basic building blocks of graphs

The flow of association and causation

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Graph terminology




Graph terminology

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Graph terminology: not a graph

4Graph terminology

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x

y


4Graph terminology

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y


4Graph terminology

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x

y


4Graph terminology

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Graph terminology: Terminology Machine Gun

5

termterm

termtermterm

termtermterm

Graph terminology

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Graph terminology

6Graph terminology

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Graph terminology

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A B

C D

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Graph terminology

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Graph terminology

6

NodesA B

C D


Node Node

Node Node

Graph terminology

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Graph terminology

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A B

C D


Edge

Edge

Edge

Edg

eEdges

Graph terminology

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Graph terminology

6

A B

C D


UndirectedGraph

Graph terminology

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Graph terminology

6

DirectedGraph

A B

C D

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Graph terminology

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Graph terminology

6

DirectedGraph

A B

C D


Parent Child

Graph terminology

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Graph terminology

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A B

C D


Parent Child

Adjacent

Graph terminology

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Graph terminology

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A B

C D


NotAdjacent

Graph terminology

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Graph terminology

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A B

C D


Path

Graph terminology

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Graph terminology

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A B

C D


Path

Graph terminology

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Graph terminology

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A B

C D


DirectedPath

Graph terminology

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Graph terminology

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A B

C D


Ancestor Descendant

DescendantDescendant

Graph terminology

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Graph terminology

6

A B

C D

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Graph terminology

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Graph terminology

6

A B

C D

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Cycle

Graph terminology

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Graph terminology

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A B

C D


Graph terminology

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Graph terminology

6

A B

C D


Directed Acyclic Graph (DAG)

Graph terminology

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Graph terminology

6


A B

C D

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Graph terminology

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Graph terminology

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A B

C D

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Immorality

Graph terminology

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Graph terminology





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Naively modeling the joint distributionStatistical modeling (no causality):

8Bayesian networks and causal graphs

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8

P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)

<latexit sha1_base64="UyTyA1Kdh3gJGoeUrvSEJndpAjw=">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</latexit>


Brady Neal / 35


8

P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)



Brady Neal / 35

P (x1, x2, x3, x4) = P (x1)P (x2 | x1)P (x3 | x2, x1)P (x4 | x3, x2, x1)

<latexit sha1_base64="o+p/T7L1m/dcl7x2u6T1WnR8yhE=">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</latexit>


8

P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)



Brady Neal / 35

x1 x2 x3 P (x4 | x3, x2, x1)

0 0 0 ↵1

0 0 1 ↵2

0 1 0 ↵3

0 1 1 ↵4

1 0 0 ↵5

1 0 1 ↵6

1 1 0 ↵7

1 1 1 ↵8

<latexit sha1_base64="m/v+zV2Ka3XPHK+QAAqANiDr0iw=">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</latexit>




8

P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)



Brady Neal / 35

x1 x2 x3 P (x4 | x3, x2, x1)

0 0 0 ↵1

0 0 1 ↵2

0 1 0 ↵3

0 1 1 ↵4

1 0 0 ↵5

1 0 1 ↵6

1 1 0 ↵7

1 1 1 ↵8





8

P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


2n�1

<latexit sha1_base64="+9ySUnU6KlrKQKB0EV5LG7T9HnY=">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</latexit>

parameters!


Brady Neal / 35

x1 x2 x3 P (x4 | x3, x2, x1)

0 0 0 ↵1

0 0 1 ↵2

0 1 0 ↵3

0 1 1 ↵4

1 0 0 ↵5

1 0 1 ↵6

1 1 0 ↵7

1 1 1 ↵8





8

P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


2n�1


parameters!

X1 X2

X3 X4

<latexit sha1_base64="XWIFaVq5D6n8XxXfE9BPk69Sd8E=">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</latexit>


Brady Neal / 35

x1 x2 x3 P (x4 | x3, x2, x1)

0 0 0 ↵1

0 0 1 ↵2

0 1 0 ↵3

0 1 1 ↵4

1 0 0 ↵5

1 0 1 ↵6

1 1 0 ↵7

1 1 1 ↵8





8

P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


P (x1, x2, . . . , xn) = P (x1)Y

i

P (xi | xi�1, . . . , x1)


2n�1


parameters!

X1 X2

X3 X4



Brady Neal / 35

Local Markov assumptionGiven its parents in the DAG, a node X is independent of all of its non-descendants.


Brady Neal / 35


9

X1 X2

X3 X4







Brady Neal / 35


9

X1 X2

X3 X4




P (x1, x2, x3, x4) = P (x1)P (x2 | x1)P (x3 | x2, x1)P (x4 | x3)

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Brady Neal / 35


9





X1 X2

X3 X4

<latexit sha1_base64="XmBsWAmtXw3NDfG6PRb34AM0Nb8=">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</latexit>


Brady Neal / 35


9





X1 X2

X3 X4


Question:How will the factorization change now?


Brady Neal / 35


9

P (x1, x2, x3, x4) = P (x1)P (x2)P (x3 | x1)P (x4 | x3)

<latexit sha1_base64="kMBmF47uyoLCL6sp+Td+RBLHARM=">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</latexit>

X1 X2

X3 X4


Question:How will the factorization change now?


Brady Neal / 35

Bayesian network factorization


Brady Neal / 35


10

P (x1, . . . , xn) =Y

i

P (xi | pai)

<latexit sha1_base64="Pa7R4s6Fv20mxBNTW+wcVpyCDVE=">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</latexit>


Brady Neal / 35


10

P (x1, . . . , xn) =Y

i

P (xi | pai)


local Markov assumption Bayesian network factorization =)

<latexit sha1_base64="WuvX0VO9o9dAwOcsTKhIQj9C2aw=">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</latexit>


Brady Neal / 35


10

P (x1, . . . , xn) =Y

i

P (xi | pai)







Brady Neal / 35


10

P (x1, . . . , xn) =Y

i

P (xi | pai)






See Chapter 3 of Koller & Friedman (2009) book for proofs


https://mitpress.mit.edu/books/probabilistic-graphical-models

Brady Neal / 35

Minimality assumption


Brady Neal / 35

Minimality assumption1. Given its parents in the DAG, a node X is independent of all its non-

descendants (local Markov assumption).


Brady Neal / 35



11

X Y

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Brady Neal / 35



11

P (x, y) = P (x)P (y | x)

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Permits distributions where

X Y



Brady Neal / 35



11

P (x, y) = P (x)P (y | x)


P (x, y) = P (x)P (y)

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X Y


and also where


Brady Neal / 35


descendants (local Markov assumption).2. Adjacent nodes in the DAG are dependent.

11

P (x, y) = P (x)P (y | x)


P (x, y) = P (x)P (y)



X Y


and also where


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11

P (x, y) = P (x)P (y | x)


P (x, y) = P (x)P (y)



X Y


and also where


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11

P (x, y) = P (x)P (y | x)


P (x, y) = P (x)P (y)



X Y


and also where

X Y

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Assumptions flowchart

12

StatisticalIndependencies

StatisticalDependencies

MarkovAssumption

MinimalityAssumption

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Recall:1. How is the local Markov assumption related to the

Bayesian network factorization?2. What are the two parts of the minimality

assumption? What do we gain with the second part?

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What is a cause?

A variable X is said to be a cause of a variable Y if Y can change in response to changes in X.


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Causal edges assumption

15

In a directed graph, every parent is a direct cause of all its children.


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16



CausalDependencies

MarkovAssumption


Causal EdgesAssumption

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Brady Neal / 35


16



CausalDependencies

MarkovAssumption



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Two assumptions to give us flow of association and causation in graphs:1. Markov Assumption2. Causal Edges Assumption

DAG +


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Graph terminology





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Graphical building blocks

18The basic building blocks of graphs

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18

X1 X2

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Two nodes:


Brady Neal / 35


18

X1 X2

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X1 X2


Two nodes: or


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18

Chain

X1 X2 X3

<latexit sha1_base64="gdBRLR1cDnTJHmNS4mSpfP1KcKg=">AAAJxnicfVVtTyM3EE6uL3fQt6P90A/94haQQFrSJCDR6hTpJOCEqruKk3hrSYS8Xm9i4TfZXkKwtuqv6C/o1/b/9N907CyQLOQ2ijSe55nxeGY8TjVn1rXb/zWfffTxJ58+f7G0/NnnX3z51cuVr0+tKgyhJ0RxZc5TbClnkp445jg914ZikXJ6ll7tBfzsmhrLlDx2E00HAg8lyxnBDlSXK81v+ykdMukdu7rVjLjC0HIZVd86+lVl1N6v+xKWF9ZhRwdo47yzibBDG+2kvYn82vllZ618tYjb3UQXhg1HDvVUjs47g2jR/YDF9rxFd2qxHSxmAjzIhrMBauxG08goAHHfV4/Q7j26vVk561OZzeXg8uVqu9WOH3osdCphtVF9R5crz//qZ4oUgkpHOLb2otPWbuCxcYxwSGq/sFRjcoWH9AJEiQW1Ax9LWKJ10GQoVwb+0qGonbXwWFg7ESkwBRzC1rGgfAq7KFz+08AzqQtHJZlulBccOYVCP6CMGUocn4CAiWEQKyIjbDBx0DXLc9ukopxfK3XlcGrLUIc3ELpkhCLQcDof300O/uZZQRM6ELZYR8eQd1SR5y2n2ZlThSKV91aWOsfk0CKlHRPstkoiwYXFHA0N1iPbAvIvhQ1Z1ZMtja2DKEfMwpbGRd/BJ2epwWbi7QhrapOMEmXiJbEJNkaNbUIwJ5XcEtThJGcu0cqywIIgIM7gCNz52FNbb6GTbxJcOJWE1oYUQ3NLOH2vg4hAWGYoCImlgkFA5CqJdvEG/NiybsKB6innTFuaoMzgcYIEk0wUAo1ZBs3cg57cBR/l1FQrJt29KfKEGei9O1MmJTWQMt2DRt6BjXPG+TQ0Al0HjWt7vqxcpWzaGjSr/PV8PE91qgzbEc0qLuUzez7s0tUuQZizoexxmoNsudJgtBzrtwclUQJ2FgIyAXWXdFwtfP+g9P3QzGnqD8pyHjvFpkKN8GFRw5nMqAaGpkaj/vfhh+KixpN3RKkckD5Mn3qFKQFZAtdGOchNDEJjDk1IH5QPzOMqihBhRvNZZLWz2q0OQbnvG461XwvKtbIvrqiRXVH4sC7rx9tX4JXlQoS6RQ/M+UyVfcotKKA4rlLkrGaq8UPiQK6hGX1AQa6hRF1jw6D2tvTnNczBk+PC4Cv9cQ1ShQMZPP/2yJ/MVQG5MODwrAYKCuPIwdX37+r+3Iia69ABvy8M8Brz0t8sijGi7ukwIzZZHGnExwuCjaBYEG8Eb2sgtTC0QCr9+3qlSn8EtyRckzACbewfeOPjQPRajaGrwl3fMnCf/OHxu7el39ve6Wy/KRdSU17QO253b3d/v7uYq2HUbk2nb2VxsBt+EFM/qqHn/SzpadsZMmQxu5uUd0Y8PO8wpCc1Ykz3DK923hkyvLOKsDinF5HhJe/U3+3Hwmm31dlp/fy+u/p6p3rTXzS+a/zQ2Gh0GruN143DxlHjpEGafzT/bv7T/HfpcEkuFUvjKfVZs7L5pjH3Lf35P9nxa7U=</latexit>

X1 X2


X1 X2


Two nodes: or


Brady Neal / 35


18

Chain Fork

X2

X1 X3

<latexit sha1_base64="PKsa4I9CLOTSgVPbhmIY41Iw7z8=">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</latexit>

X1 X2 X3


X1 X2


X1 X2


Two nodes: or


Brady Neal / 35


18

Chain Fork

X1

X2

X3

<latexit sha1_base64="j56l+BkZBCAkQQBD62a1TQy/gIA=">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</latexit>

X2

X1 X3

<latexit sha1_base64="PKsa4I9CLOTSgVPbhmIY41Iw7z8=">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</latexit>

X1 X2 X3


X1 X2


X1 X2


Immorality

Two nodes: or


Question:What assumption tells us that X1 and X2are associated, given the following graph?

X1 X2

<latexit sha1_base64="OFku4N065IrcmY6PeUpgE9VCCFw=">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</latexit>

Brady Neal / 35

Chains and forks: dependence

20

X1 X2 X3

<latexit sha1_base64="p+paP62jjcy8I1BzOr/2ycZSPls=">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</latexit>


Brady Neal / 35


20

X1 X2 X3

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association


Brady Neal / 35


20

X1 X2 X3


X2

X1 X3

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association


Brady Neal / 35


20

X1 X2 X3


X2

X1 X3

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association

association


Brady Neal / 35

Chains and forks: independence

21

X1 X2 X3


X2

X1 X3


association

association


Brady Neal / 35

X2X1 X3

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21

X2

X1 X3


association

association


Brady Neal / 35

X2

X1 X3

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X2X1 X3



21

association

association


Brady Neal / 35

X2

X1 X3

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X2X1 X3



21

association

association

blocked path blocked path


Brady Neal / 35


21

X1 X2 X3


X2

X1 X3


association

association

unblocked path unblocked path


Brady Neal / 35

Proof of conditional independence in chains

22

X2X1 X3



Brady Neal / 35

Proof of conditional independence in chainsGoal: show

22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)

<latexit sha1_base64="QXPWHG5I0Q7gPP2c6TI9cihyOwg=">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</latexit>


Brady Neal / 35


22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)


1. Bayesian network factorization:


Brady Neal / 35


22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)


1. Bayesian network factorization: P (x1, x2, x3) = P (x1)P (x2|x1)P (x3|x2)

<latexit sha1_base64="AaufpYlP1bhSK5zcmx1aCuqkgFw=">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</latexit>


Brady Neal / 35


22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)



2. Apply Bayes’ rule:

P (x1, x2, x3) = P (x1)P (x2|x1)P (x3|x2)


P (x1, x3 | x2) =P (x1)P (x2|x1)P (x3|x2)

P (x2)

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Brady Neal / 35


22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)




P (x1, x2, x3) = P (x1)P (x2|x1)P (x3|x2)


P (x1, x3 | x2) =P (x1)P (x2|x1)P (x3|x2)

P (x2)



Brady Neal / 35


22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)




P (x1, x2, x3) = P (x1)P (x2|x1)P (x3|x2)


P (x1, x3 | x2) =P (x1)P (x2|x1)P (x3|x2)

P (x2)



Brady Neal / 35


22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)




3. Apply Bayes’ rule again:

P (x1, x2, x3) = P (x1)P (x2|x1)P (x3|x2)


P (x1, x3 | x2) =P (x1)P (x2|x1)P (x3|x2)

P (x2)



Brady Neal / 35


22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)





P (x1, x2, x3) = P (x1)P (x2|x1)P (x3|x2)


P (x1, x3 | x2) =P (x1)P (x2|x1)P (x3|x2)

P (x2)


P (x1, x3 | x2) =P (x1, x2)

P (x2)P (x3|x2)

= P (x1|x2)P (x3|x2)

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Brady Neal / 35


22

X2X1 X3


P (x1, x3 | x2) = P (x1 | x2)P (x3 | x2)





P (x1, x2, x3) = P (x1)P (x2|x1)P (x3|x2)


P (x1, x3 | x2) =P (x1)P (x2|x1)P (x3|x2)

P (x2)


P (x1, x3 | x2) =P (x1, x2)

P (x2)P (x3|x2)

= P (x1|x2)P (x3|x2)


P (x1, x3 | x2) =P (x1, x2)

P (x2)P (x3|x2)

= P (x1|x2)P (x3|x2)



Brady Neal / 35

Proof of conditional independence in forks

Your turn J


Brady Neal / 35

Immoralities

24

X1

X2

X3

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Brady Neal / 35

Immoralities

24

X2

X1 X3

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blocked path


Brady Neal / 35

Immoralities

24

X2

X1 X3


collider

blocked path


Brady Neal / 35

Immoralities

24

X2

X1 X3


collider

P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)

<latexit sha1_base64="tkR/YMsVXeyJaupzva80mAvAn2U=">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</latexit>

blocked path


Brady Neal / 35

Immoralities

24

X2

X1 X3


collider

P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)


blocked path

P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)



Brady Neal / 35

Immoralities

24

X2

X1 X3


collider

P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)


blocked path

P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)


P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)



Brady Neal / 35

Immoralities

24

X2

X1 X3


collider

P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)


blocked path

P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)


P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)


P (x1, x3) =X

x2

P (x1, x2, x3)

=X

x2

P (x1)P (x3)P (x2 | x1, x3)

= P (x1)P (x3)X

x2

P (x2 | x1, x3)

= P (x1)P (x3)



Brady Neal / 35

Immoralities: conditioning on the collider

25

X2

X1 X3


blocked path


Brady Neal / 35

Immoralities: conditioning on the collider

25

X2

X1 X3

<latexit sha1_base64="50asm3XYjk2lTGNJ0xVqCIasUKc=">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</latexit>

unblocked path


Brady Neal / 35

Example: good-looking men are jerks

26

X2

X1 X3



Brady Neal / 35


26

X1 =

(1 good-looking

0 otherwise

<latexit sha1_base64="HeFX0zcVezyxX4BVeiXhloxAgiM=">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</latexit>

X2

X1 X3



Brady Neal / 35


26

X1 =

(1 good-looking

0 otherwise


X3 =

(1 kind

0 jerk

<latexit sha1_base64="f6URPxn2+0hb77qn/qcxhcPTVPU=">AAAJAHicfVVLbxs3EN6kj6juy2mPvbB1XPQgK5JswOlBQADbQVAkgFP41WYNg8sdWaz4Asm1LBN76Y/ob+it6K3oP+ml1/ZndLha29La6goChvN98+BwOMyM4M53u389ePjOu++9/6j1wcqHH338yaerjz87crqwDA6ZFtqeZNSB4AoOPfcCTowFKjMBx9l4J+LHF2Ad1+rATw2cSnqu+JAz6lF1tvr9ydkmGZA0g3OuAkNPriQrpPp65GuSerj0YcxVXpI0rYHuDfAT2HEZtSmovDY/W13rdrrVR+4KvVpYS+pv/+zxo1/SXLNCgvJMUOfe9rrGnwZqPWcCypW0cGAoG9NzeIuiohLcaag2X5J11ORkqC3+lSeVdt4iUOncVGbIlNSPXBOLyvuwt4UfPjsNXJnCg2KzQMNCEK9JrCTJuQXmxRQFyizHXAkbUUuZx3qvLITJZLm41nrsaYalJuvkBaauOAOCGgGL+V0O0d8iK2ri2WGIdXLAx1ekJi9azqqzoPJILm+sHHjP1bkj2ngu+VVdREYLRwU5t9SMXAfJ3xUuVtVMNwx1HrMccYchra98R5+CZ5baaXAjasC1c2DaVu3l2tRaPXFtRgWr5Y4ET9tD7ttGOx5ZmATmGR2hu1C12MYriu3VpoXXbaXzWGvnqcLdD3qESUJVTqLQdiA5JsTG7coOSR6edpyfCqQGEIIbB22SWzppE8kVl4UkE577ETZ9t7ONPsqZqdFc+RtTEhi32HvXplwpsFgyM8BG3sLAQy7ELDWGXYeN6wahrF1lfNYakNf+BqHaT72rnLoR5DUXxFzM2yh949uECn6uBgKGKDuhDRqtVOe3g0eiJUaWEiuB565gUi9CuleGNDZzloW9slzEjqitUStDXDRwvOZgkGHAGpJ+GX+kWjR46pqotEfS/9NnXnE+YJXQtdUea1MlYajAJoRb5S3zoM4iZpjDcB5Z6631602ACKkV1IQnUfmkTOUYrOrLIsR12dzerkavfChlPLfKA/ch12UKwsFsotWKIW+YGnpbOJQbaA63KMoNlOkLajmevSvDSQPzOKx9HHxlOGhAuvAoo+cf7vhTQ11gLSw6PG6AEnAcebz64XXTnx+BvYgd8OPSBC+oKMPlshwr1N+fZoVNl2da4ZMlyVagXJJvBV41QHA4tFAqw5vmSZVhH29JvCZxBLqqf/B1rAZiMHqCXRXv+obF+xReHrx+VYadza3e5otyKTUTBVxz+zvbu7v95VyDo3ZjNn1ri73t+MOc0kqNPR/mSffbzpGxivn1pLw2wtEw8jikpw1iVe45XmO/c2R8ZzXj1ZxeRsaXvNd8t+8KR/1Ob6vz7ZuttefP6je9lXyRfJV8k/SS7eR58jLZTw4TlvyR/J38k/zb+rn1a+u31u8z6sMHtc3nycLX+vM/lCs61Q==</latexit>

X2

X1 X3



Brady Neal / 35


26

X1 =

(1 good-looking

0 otherwise


X3 =

(1 kind

0 jerk


X2 = X1 AND X3 =

(1 in relationship

0 not in relationship

<latexit sha1_base64="TZcfoNx3Vt2ujwp6lKqBUSpyuHM=">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</latexit>

X2

X1 X3


X2 = X1 AND X3 =

(1 in relationship




Brady Neal / 35


26

X1 =

(1 good-looking

0 otherwise


X3 =

(1 kind

0 jerk


X2 = X1 AND X3 =

(1 in relationship



X2 = X1 AND X3 =

(1 in relationship



X2

X1 X3


X2 = X1 AND X3 =

(1 in relationship




Brady Neal / 35

Good-looking men are jerks scatterplot


looks

kind

ness

Full population

Brady Neal / 35



looks

kind

ness

Groups by availability

Brady Neal / 35



looks

kind

ness

Available men

Brady Neal / 35

Conditioning on descendants of colliders

28

X1

X2

X3

X4

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X1 6?? X3 | X4

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Brady Neal / 35


28

X1

X2

X3

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X1 6?? X3 | X4

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Question:In the three different kinds of three-node graphs, what can block a path?

Brady Neal / 3530

Graph terminology





Brady Neal / 35

Blocked path definition

31The flow of association and causation

Brady Neal / 35


31

A path between nodes X and Y is blocked by a (potentially empty) conditioning set Z if either of the following is true:


Brady Neal / 35


31

A path between nodes X and Y is blocked by a (potentially empty) conditioning set Z if either of the following is true:1. Along the path, there is a chain or a fork

where is conditioned on ( ).W

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W 2 Z

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· · · W ! · · ·

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· · · ! W ! · · ·

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where is conditioned on ( ).2. There is a collider on the path that is not conditioned on ( )

and none of its descendants are conditioned on ( ).

W


W 2 Z


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· · · ! W ! · · ·


W 62 Z

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de(W ) 6✓ Z

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W



Brady Neal / 35


31


where is conditioned on ( ).2. There is a collider on the path that is not conditioned on ( )

and none of its descendants are conditioned on ( ).

W


W 2 Z


· · · W ! · · ·


· · · ! W ! · · ·


W 62 Z

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de(W ) 6✓ Z

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W


Unblocked path: a path that is not blocked


Brady Neal / 35

d-separation


Brady Neal / 35

d-separationTwo (sets of) nodes X and Y are d-separated by a set of nodes Z if all of the paths between (any node in) X and (any node in) Y are blocked by Z.


Brady Neal / 35

X ??G Y | Z =) X ??P Y | Z

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32

Theorem: Given that is Markov with respect to ,P

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G

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Brady Neal / 35

X ??G Y | Z =) X ??P Y | Z



32



G


global Markov assumption


Brady Neal / 35

X ??G Y | Z =) X ??P Y | Z



32



G


global Markov assumptionlocal Markov assumption ()

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Brady Neal / 35

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global Markov assumptionlocal Markov assumption ()

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Markov assumption


Brady Neal / 35

d-separation practice

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Brady Neal / 35


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Brady Neal / 35


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Brady Neal / 35


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Brady Neal / 35


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Brady Neal / 35


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Brady Neal / 35


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Brady Neal / 35


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Brady Neal / 35


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<latexit sha1_base64="SAr9QcxK+++XMCbga9rmD03o0Qo=">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</latexit>



CausalDependencies

MarkovAssumption



<latexit sha1_base64="BvLrR0JmMb8EBOlrA/pb0a+2+Sw=">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</latexit>


Brady Neal / 35


34

X

T Y

<latexit sha1_base64="no+7bwXz+dv7ufugW+w3uBmazK4=">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</latexit>



CausalDependencies

MarkovAssumption




association

association


Brady Neal / 35


34

X

T Y

<latexit sha1_base64="no+7bwXz+dv7ufugW+w3uBmazK4=">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</latexit>



CausalDependencies

MarkovAssumption




association

association


Brady Neal / 35

causal association(causation)


34

X

T Y

<latexit sha1_base64="5hv7P805kKHR+YrUbnkAzXqWANo=">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</latexit>



CausalDependencies

MarkovAssumption




association


Brady Neal / 35

confounding association

causal association(causation)


34

X

T Y

<latexit sha1_base64="5hv7P805kKHR+YrUbnkAzXqWANo=">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</latexit>



CausalDependencies

MarkovAssumption





Brady Neal / 3535

T M1 M2 Y

W1

W2

W3

X3

X2

X1

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Brady Neal / 35

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Brady Neal / 3535

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causal association


Brady Neal / 3535

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causal association

non-causal association

non-causal association


Documents

The Flow of Association and Causation in Graphs