Causal&Inference:&predicon,& explanaon,&and&interven1on& · Causal&Inference:&predicon,&...

Causal Inference: predic1on, explana1on, and interven1on

Lecture 5: Causality and Graphical Models

Samantha Kleinberg [email protected]

Next few weeks

•  October 13 (Tuesday class!): Time series •  October 19: Lecture + midterm review/Q&A •  October 26: Midterm exam •  November 2: Project proposal due

•  November 30 and December 7: final presenta1ons

Final Project

•  Project types (not exhaus1ve!) – Analyze data, adapt causal inference method to par1cular domain

–  Compare causal inference methods –  Theore1cal work on causality (methods or meaning)

•  Proposal – What’s the goal? How will you accomplish it? How will you know the project was successful? What resources are needed (and do you have them)? 1 page

Today

•  What makes a graphical model causal? •  How to use graphical models to answer causal ques1ons – Predic1ng effects of ac1ons – Counterfactual queries – Explana1on

Applica1on: infant mortality

Mani, S., & Cooper, G. F. (2004). Causal Discovery Using a Bayesian Local Causal Discovery Algorithm. Proceedings of MedInfo, 731-‐735

Applica1on: epidemiology

Twardy, C. R., Nicholson, A. E., Korb, K. B., & McNeil, J. (2006). Epidemiological data mining of cardiovascular bayesian networks. Electronic Journal of Health Informa8cs, 1(1), e3

Applica1on: Organizing evidence

Lagnado, D. (2011). Thinking about Evidence. In Proceedings of the Bri8sh Academy (Vol. 171, pp. 183-‐223)

Applica1on: Causes and effects of poverty

Bessler, D. A. (2003). On world poverty: Its causes and effects. Food and Agricultural Organiza8on (FAO) of the United Na8ons, Research Bulle8n, Rome

Applica1ons of causal BNs

•  Biomedical informa1cs – diagnosis, prognosis •  Psychology – representa1on of causal learning •  Economics/finance

Markov condi1on

(from last week) Node independent of non-‐descendants given its parents

X

Y Z

Y ⊥ Z | X

d-‐separa1on

X Y Z

DAG G Set of independencies

Y ⊥ Z | X

d-‐separa1on

d-‐separa1on

Equivalent statements, for sets of nodes X, Y, Z in graph G: – X and Y are d-‐separated by Z (Z can be node or set of nodes) in G

– X and Y are condi1onally independent given Z – Z blocks all paths between X and Y

d-‐separa1on and Markov blanket

Markov blanket: set of nodes that separate a node from all others d-‐separa3on: Method for determining whether a pair of nodes (or sets of nodes) are independent condi1oned on another set

Defini1on: d-‐separa1on •  Node v is a collider if two arrowheads meet at v

•  X and Y are d-‐connected by Z in graph G iff –  Exists an undirected path between a vertex in X and vertex in Y s.t. for every collider C on the path, C or descendant of C is in Z and no non-‐collider on path is in Z

•  X and Y are d-‐separated by Z in G iff they are not d-‐connected by Z in G

Z Y X Z Y X

✓ ✗

Example 1

•  X → Y → Z

•  X ß Y → Z

In both cases, X,Z d-‐separated by Y: no colliders on path from X to Z, and X and Z not d-‐connected by Y

Example 2

•  X → Y ← Z

•  X,Z d-‐separated by a set of nodes only if Y NOT in that set. X,Z d-‐connected by Y

Example 3

•  Are Y,Z d-‐separated by W?

•  No, d-‐connected by X and W is descendent

X

Y Z

W

But…

Indep -‐> many networks Network -‐> 1 set indep

X Y Z Y ⊥ Z | X

X Y Z

X

Y Z

But…

Can rule out some

X Y Z Y ⊥ Z | X

X Y Z

X

Y Z X

Y Z

X

…And

HDL

Heart disease

Genes

Heart disease HDL

…Also Coin 1 Coin 2 # obs.

H H 5

T T 3

H T 1

T H 1

P(C1 = H ^C2 = H )> P(C1 = H )P(C2 = H )5 /10 > 6 /10*6 /10

C1 /⊥C2

Causal interpreta1on

•  Causal Markov condi1on •  Faithfulness •  Causal sufficiency

+ a few others, e.g. variables “correctly” specified

Causal graph

•  Arrows denote direct causes – Edge from X to Y means X causes Y

•  DAG

ads buy weather

Causal Markov condi1on (CMC)

Node in the graph is independent of all of its non-‐descendants (direct and indirect effects) given its direct causes

CMC and screening off

Recall Common Cause Principle (CCP) If P(X^Y)>P(X)P(Y) then either X causes Y (or vice versa) or they have a common cause

Now: if P(X^Y)>P(X)P(Y) and they have a common cause C, it means X ind Y | C Note that CCP seeks single common cause. CMC allows for sets of nodes.

Problems: feedback

Supply

Demand

Problems: Indeterminism

P(picture|switch) < P(picture|switch,sound)

TV switch

Picture Sound

TV switch

Picture Sound

Closed Circuit

Spirtes, P., Glymour, C., & Scheines, R. (2000). Causa8on, predic8on, and search. MIT Press

Problems: hidden common causes

CFS

Tired Achy

Flu

Completeness of graph

•  Complete: all common causes included, all causal rela1ons among variables included

•  Incomplete: not all intermediate factors necessarily included

Faithfulness

Exactly the dependencies in the underlying structure hold in the data –  i.e. Independence rela1ons not from chance but from structure

– No canceling out

Example

Smoking

Exercise

Health

-‐ +

+

Another example

Gene A

Gene B Phenotype

-‐ +

+

A final example (determinis1c chain)

X Y Z

X ⊥ Z |Y

Selec1on bias

fever Abdominal pain F+ A

Go to hospital Stay home

Cooper, G. F. (1999). An overview of the representa1on and discovery of causal rela1onships using bayesian networks. In C. Glymour & G. F. Cooper (Eds.), Computa8on, causa8on, and discovery. AAAI Press and MIT Press

Recap of problems for faithfulness

•  Only true in large sample limit •  Simpson’s paradox •  Selec1on bias •  Sta1s1cal tests

Quick recap

•  CMC: popula1on produced by structure has these independencies

•  Faithfulness: popula1on has only these independencies Why do we need both?

X Y Z

Causal sufficiency

•  All common causes of pairs of variables measured

•  Not sufficient if Y not measured

X

Y

Z

Completeness vs. sufficiency

Completeness: common causes are included in causal graph Sufficiency: all common causes have been measured

Example

Intelligence

X Y Z

Scheines, R. (1997). An introduc1on to causal inference. Causality in Crisis, 185-‐99

What if no causal sufficiency?

•  Need to include all graphs with unmeasured common causes.

•  Ex: measured A, B, C. Found A ╨ C. With CMC, faithfulness but no causal sufficiency, the following graphs are all possible (X is unmeasured):

A

B

C A

B

C

X

A

B

C

X . . .

In absence of sufficiency…

•  Can s1ll learn something – Some rela1onships may appear in all graphs – Can find set of all graphs represen1ng independence rela1ons, with nodes for possible hidden variables

•  Timing informa1on helps

Things to beware of with inference

•  Sample size •  Missing data (not just variables) •  Mul1ple tes1ng (and FDR) •  What structures DAG can/cannot represent (e.g. 1me series and feedback)

•  Variable representa1on

The good news

•  Can add 1me •  Can experiment •  Methods for tes1ng assump1ons

Recap of causal inference with BN

What makes a Bayesian network causal? The assump1ons: CMC, sufficiency, faithfulness

Assump1ons+DataàIndependencies à Causal BN(s) à effects of interven1ons

Learning BN

Same methods we discussed last week – Search and score – Constraint based (e.g. PC algorithm)

Pearl on causality

•  Ac1ons – What happens if we do X?

•  Counterfactuals – What if things happened differently?

•  Explana1ons – Why did X happen?

Manipulability

BPA

Obesity

Ideal manipula1ons

•  Defini1on: change in value of a variable that does not introduce any other changes (except those produced by the change in variable)

Weather

Running speed Air condi1oner

Speeches

Popularity Dona1ons

Tes1ng popularity, how do we manipulate it’s value?

Seeing versus doing

Disease

Treatment

Doctor

Outcome

Hospital

Predic1ng the effects of ac1ons

What’s P(C) if I turn the sprinkler on? Is this the same as P(C|S=T)?

Cloudy

Sprinkler Rain

Wet grass

Interven1on and joint probability

!= just incorpora1ng evidence – Last week: set value of observed values – This week: set value by forcing variable to take value independent of its parents’ values

Cloudy

Sprinkler Rain

Wet grass

If turn on sprinkler, the fact that it’s on no longer gives info about C

Interven1on and joint probability

Cloudy

Sprinkler Rain

Wet grass

P(C,S,W,R) = P(c)P(s | c)P(r | c)P(w | s, r)C,S,W ,R∑

P(C,W,R | do(s)) = P(c)P(s)P(r | c)P(w | s, r)C,W ,R∑

do() operator

Model can help us determine the effect of interven1ons P(X=x|Y=y) != P(X=x|set Y=y)

Example

P(S|do(M))

Disease

Medica1on

Side effect

do-‐calculus rules

1.  Inser1on/dele1on of observa1ons

P(y | do(x), z,w) = P(y | do(x),w) if (Y⊥ Z|X,W)GX

Disease

Medica1on

Side effect

GX Means edges into X deleted

do-‐calculus rule 1 examples

1.  Inser1on/dele1on of observa1ons

P(y | do(x), z,w) = P(y | do(x),w) if (Y⊥ Z|X,W)GX

D

M

S

G GS

D

M

S

X

W

Y

Z

G

X

W

Y

Z

GX


2. Ac1on/Observa1on exchange If remove edges from Z, and independent, can replace doing with observing

P(y | do(x),do(z),w) = P(y | do(x), z,w) if (Y⊥ Z | X,W)GXZ

X Y Z

W

X Y Z

W

G G-‐XZ


3. Inser1on/dele1on of ac1ons Z(W) is set of Z nodes that are not ancestors of W-‐nodes in

P(y | do(x),do(z),w) = P(y | do(x),w) if (Y ⊥ Z | X,W )GX,Z (W )

X Y Q

W

Z X Y Q

W

Z

GX

G G –X,Z(W)

Summary of do-‐calculus

1.  Inser1on/dele1on of observa1ons 2.  Ac1on/Observa1on exchange 3.  Inser1on/dele1on of ac1ons

In general, may have unobserved/hidden variables

Some caveats

•  Time •  Modularity •  Possibility of intervening •  Efficacy

Counterfactuals reminder

•  If I had not gone running, I would not have go}en a sunburn

•  If the pa1ent had taken the drug, she would have recovered

•  Had I bought shares of Apple stock in 2004, I would have made a large profit

Pearl on Counterfactuals

Like do(), except backward looking and changing value of variable Three steps

1.  Abduc1on: use evidence to interpret past

2.  Ac1on: change to hypothe1cal values

3.  Predic1on: see consequences of ac1ons

Disease (D)

Medica1on (M)

Side effect (S)

Example from Pearl If D, then D would s1ll be true if A were false DàD¬A

Court (U)

Captain (C)

A B

Death (D)

Abduc1on

Captain (C)

A B

Death (D)

Ac1on

Court (U)

Captain (C)

A B

Death (D)

Predic1on

Court (U)


Actual causality

Pearl: token cause = actual cause – What caused a person’s lung cancer? – Who is responsible for an accident

Graphical models and explana1on

•  Graphical model that represents rela1onships between variables

•  Observa1ons give truth value of variables in par1cular scenario

•  Use model to evaluate counterfactual queries

Pearl’s approach to actual cause

•  Based on but a}empts to solve problems with counterfactuals – Bob and Susie and the broken bo}le

•  Key idea: sustenance (mix of necessity and sufficiency) –  If Bob’s rock missed, would Susie’s sustain the glass breaking?

Defini1ons

•  Depends (necessity) –  y depends on x, if x is necessary to maintain value of y

•  Produces (sufficiency) –  x can produce y if it can bring about effect when neither are present

•  Sustains –  x sustains y if there is at least one condi1on where Y will differ from y in absence of x AND Y=y is maintained in presence of x under any set of condi1ons

Causal Beams

•  Causal beam: New model, where we remove all parents except those that minimally sustain their children. Set other parents to some w’.

•  x is actual cause of y if x is necessary for y in that causal beam for some w’

Example of causal beam

•  Did the traveler die of thirst or poisoning?

•  Death = C v D

X (enemy2 shoots canteen)

D (dehydra1on)

p(enemy1 Poisons water)

C (cyanide Intake)

Y (death)


Construc1ng the causal beam

•  True: X,D,Y,P •  False: C

•  Note: C=¬X^P

X=1

D =1

P=1

C =0

Y =1

X= shoots canteen, D=dehydra1on, Y=Death, C=cyanide intake, P=poisons water

Sustaining

Inac1ve



•  Now: C=¬X

X=1

D =1

P=1

C =0

Y =1


Sustaining



•  Next: Y=DvC

X=1

D =1

P=1

C =0

Y =1


Sustaining

Sustaining Inac1ve



•  Finally: Y=D, Y=X

X=1

D =1

P=1

C =0

Y =1


Sustaining

Sustaining

What if we are uncertain?

•  U represents 1me 1ll first drink

•  U=1 if canteen emp1ed before drink

•  U=0 otherwise

X=1

D

P=1

C

Y

u

Case 1: u=1

•  Canteen emp1ed before traveler drinks

•  Same beam as before

X=1

D =1 C =0

Y =1

P=1 U=1

Case 2: u=0

•  Drinks before canteen is emp1ed

•  What if U is uncertain? •  Use P(u) to calculate •  P(x caused y)= – Sum of P(u) over u where x caused y in u

X=1

D=0

P=1

C=1

Y=1

U=0

Problem: Over-‐determina1on

•  Which rifleman caused the death?

•  Counterfactual: –  If not A, then B would have caused D

Court (U)

Captain (C)

A B

Death (D)

Problem: Over-‐determina1on

•  Which rifleman caused the death?

•  Counterfactual: –  If not A, then B would have caused D

•  Structural: – A sustains D against B

Court (U)

Captain (C)

A

Death (D)

B=False

Challenges for the actual cause

•  Type !=Token •  Subjec1vity •  Timing

Challenge: subjec1vity

•  Variables chosen and what values they are set to affect outcome – Smoking vs. dura1on of smoking

•  Note: both queries and their answers may then differ

Intoxica1on

Car Accident

Weather Emo1onal state

Challenge: 1me

•  Bob starts smoking (S) Wednesday. He’s diagnosed with lung cancer (LC) on Friday. Did his S cause his LC?

•  Bob later dies. Was LC the cause?

Smoking

Lung cancer

Death

Inference recap BN DBN Granger Temporal logic

Results Graph

Time No

Data C/D/M

Cycles No

Latent vars. Yes

Predic1on Yes

Token cause

Counterfactual -‐based

BN DBN Granger Temporal logic

Results Graph Graph Rela1onships Rela1onships

Time No Set of lags Single lag* Window

Data C/D/M C/D/M* C D/M

Cycles No Yes Yes Yes

Latent vars

Yes Yes No No

Predic1on

Yes Yes No No

Token cause

Counterfactual-‐based

No No Probabilis1c

Further reading •  Graphical models and causality –  Spirtes, P., Glymour, C., & Scheines, R. (2000). Causa8on, predic8on, and search. MIT Press

–  Pearl, J. (2000/2009). Causality: Models, reasoning, and inference. Cambridge University Press.

•  Actual cause –  Pearl’s book – Halpern, J. Y., & Pearl, J. (2005). Causes and explana1ons: A structural-‐model approach. Part I: Causes. The Bri8sh Journal for the Philosophy of Science, 56(4), 843-‐887.

So�ware

•  TETRAD V h}p://www.phil.cmu.edu/tetrad/ •  h}p://www.dagi}y.net

For next week

•  How can we find how long it takes for smoking to cause lung cancer?

•  When to buy/sell a stock a�er you hear some news?

•  Read CPT 2.4.2

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