Complete Identification Methods for the Causal Hierarchy - ClopiNet

Complete Identification
Methods for the Causal
Hierarchy
Ilya Shpitser
(joint work with Judea Pearl)
Computer Science Department
UCLA
ilyas@cs.ucla.edu

Problems We Address
How to formalize causal questions people
ask?
What sorts of assumptions are necessary to
answer causal questions people care about?
What specific assumptions are needed to
answer a given question?
Is it possible to develop a general method for
answering causal questions? If so, can
anything be proven about the generality of
this method?

Formalized Notions
Causal models (our domains)
Interventions (changes in the domain)
Causal queries:
Causal effects (responses to changes)
Counterfactuals (responses to multiple
hypothetical changes)
Answering queries from premises (causal
deduction or identification)

Graphical Causal Models
Definition: A graphical causal model is a 4-tuple
〈V,U,F,P(u)〉, where
V = {V 1 ,...,V n } are observable variables
U = {U 1 ,...,U m } are background variables
F = {f 1 ,...,f n } are functions determining V in
terms of a subset of U, V
P(u) is a distribution over U
P(u) and F induce a distribution P(v) over
observable variables

Models Induce Causal Graphs
For a model M draw a causal graph as follows:
a node for each V i ,U i
X → V i if X ∈ U ∪ V is an argument of f i
Absence of ↔ arcs between disjoint
W,Z ⊆ U implies P(W,Z) = P(W)P(Z)
Optional: drop U nodes
Genetic predisposition
Genetic predisposition
u1 u2 u1 u2 u3
Smoking
Cancer
Smoking
Tar
Cancer

Models Induce Causal Graphs
For a model M draw a causal graph as follows:
a node for each V i ,U i
X → V i if X ∈ U ∪ V is an argument of f i
Absence of ↔ arcs between disjoint
W,Z ⊆ U implies P(W,Z) = P(W)P(Z)
Optional: drop U nodes
Genetic predisposition
Genetic predisposition
Smoking
Cancer
Smoking
Tar
Cancer

Graphs and d-separation
It’s possible to reflect probabilistic independence
among variables using the graphical notion of
d-separation:
A path p is d-separated given Z if:
p contains X → W → Y , X ← W → Y , or
X ↔ W → Y , and W ∈ Z or
p contains X → W ← Y , X ↔ W ← Y , or
X ↔ W ↔ Y , and De(W) ∩ Z = ∅
X is d-separated from Y by Z (X ⊥ Y |Z) if
all paths from X to Y are d-separated by Z

Causal Queries
We consider two kinds of causal queries
Causal effects, e.g., “I have a headache.
Should I take an aspirin?”
Counterfactuals, e.g., “Would I have a
headache had I taken an aspirin?”
Crucial difference: causal effects are results
of consistent evidence, while counterfactuals
result from conflicting evidence

Interventions in Causal Models
An action do(x) sets X to the value x regardless
of the natural influences on X
The causal effect of do(x) on P(v) is an
interventional distribution P x (v) or P(v|do(x))
do(x) removes all arrows (causal influences)
incoming to X in model M to create a
submodel M x
Genetic predisposition
Genetic predisposition
Smoking = yes
Cancer
Smoking
Tar = no
Cancer

Counterfactual Events
Event Y x = y means "variable Y attains value
y under intervention do(x)" (abbreviated as y x )
We view counterfactuals as conjunctions γ of
such events: each event takes place in its
own submodel
Our example contains two events:
“no aspirin” and “headache aspirin ”
Can be represented as a distribution over
these events: P(headache aspirin |no aspirin)
(well-defined and derivable from M)

Evaluating Causal Queries
Two “direct” methods of evaluating queries:
If we know P(u) and F , we can just evaluate
We can act in our domain (or do a
randomized experiment), and measure the
outcome Y to get P(y|do(x))
Problems: P(u),F not generally known,
experiments are expensive or illegal, unclear
what to do with counterfactuals
We want to compute queries from causal
assumptions and available information

Identification
We write φ,T ⊢ id θ if θ can be uniquely
computed (identified) from φ in a class of
models described by T
φ,T ⊬ id θ if (∃M 1 ,M 2 ) ∈ T s.t. M 1 ,M 2 agree
on φ but disagree on θ
Intuition: φ are “premises,” T is the “domain
theory,” θ is the “query”
Example: mathematical logic
φ = ZF, T = set theory, θ = Axiom of Choice

Identification (cont.)
Our domain theory/description is the causal
graph (T = G)
θ is our queries (causal effects,
counterfactuals)
We consider two versions of the identification
problem:
P(v),G ⊢ id P(y|z,do(x)) (causal effects
from observations)
{P x (v \ x)|x ⊆ v},G ⊢ id P(γ|δ)
(counterfactuals from experiments)

Our Contribution
Both identification problems received some
attention in the literature, but no complete
solutions exist
Our contribution is complete algorithms for
solving these problems
Complete: failure implies all other methods
must fail
Completeness allows us to derive useful
corollaries

Corollaries
Complete graphical characterization of
identifiable causal effects
Completeness status of existing identification
algorithms (Tian’s algorithm, do-calculus)
Applications (surrogate experiments,
identifiable models)
Longer term: G induces constraints on P(v).
Which constraints are testable? Can we use
testable constraints to infer parts of G from
P(v)? (More on this in our AAAI-08 paper)

Effects: Known Graphical Criteria
Back-door criterion:
P(y|do(x)) = ∑ z
P(y|x,z)P(z) in G if
Z ∉ De(X) G
Z blocks all paths from X to Y containing an
arrow into X
U
Z
X
G
Y

Known Graphical Criteria (cont.)
Front-door criterion:
P(y|do(x)) = ∑ z P(z|x)∑ x ′ P(y|x ′ ,z)P(x ′ ) in G if
Z blocks all directed paths from X to Y
No back-door from X to Z
All back-doors from Z to Y blocked by X
U
X
Z
G
Y

Negative Graphical Criteria
Bow arc graph:
U
X
Y

Negative Graphical Criteria
Bow arc graph:
P(U) = {0.5, 0.5},X ← U
U
X
Y

Negative Graphical Criteria
Bow arc graph:
P(U) = {0.5, 0.5},X ← U
M 1 : Y ← (X + U) (mod 2),M 2 : Y ← 0
U
U
M
M
1 2
X
Y
X
Y

Negative Graphical Criteria
Bow arc graph:
P(U) = {0.5, 0.5},X ← U
M 1 : Y ← (X + U) (mod 2),M 2 : Y ← 0
Trick: bit parity is observationally the same as
a constant 0 function in the bow arc graph
U
U
M
M
1 2
X
Y
X
Y

Negative Graphical Criteria
Bow arc graph:
P(U) = {0.5, 0.5},X ← U
M 1 : Y ← (X + U) (mod 2),M 2 : Y ← 0
Trick: bit parity is observationally the same as
a constant 0 function in the bow arc graph
Conclusion: P(v),G ⊬ id P(y|do(x))
U
U
M
M
1 2
X
Y
X
Y

Identifying Effects of Singletons
Identifying P x (v \ x) (X a singleton):
Theorem (Tian 2002): P(v),G ⊬ id P x (v \ x) iff
there is a child Z of X with a bidirected path
from X to Z
Question: What about P x (y) where X,Y are
arbitrary sets? (Open since 1950s)
X
Z 1
. . .
Z Z k
. . .

do-calculus
do-calculus is a set of three rules for
manipulating interventional distributions:
1 P x (y|z,w) = P x (y|w) if Y ⊥ Z|X,W in G X
2 P x,z (y|w) = P x (y|z,w) if Y ⊥ Z|X,W in G X,Z
3 P x,z (y|w) = P x (y|w) if Y ⊥ Z|X,W in G X,Z
∗
where Z ∗ = Z \ An(W) in G X
Question: Can every identifiable effect be
derived using do-calculus? (Open since 1994)

Our Results
A complete graphical condition for
identification in the general case of P x (y)
A complete algorithm expressing P x (y) in
terms of P(v)
A proof of completeness of do-calculus for
identifying P x (y)
A characterization of models where all effects
are identifiable

C-components
C-component: maximal set of nodes pairwise
connected by bidirected paths
Given graph has two C-components:
{W,Z,M} and {X,Y }
W Z M
X
Y

General Identification Strategy
Each C-component S i in G x corresponds to a
subproblem
Key step: to test P x (y) from P(v),G, test
P v\si (s i ) for each s i , return ∑ ∏
v\x,y i P v\s i
(s i )
For example: P x (y) = ∑ w,z P z(y)P x (z,w)
W
Z
X
Y

Example (cont.)
So far: P x (y) = ∑ w,z P z(y)P x (z,w)
P z (y) = ∑ w,x
P(y|z,x,w)P(x,w) (back-door)
P x (z,w) = P(z|x,w)P(w) (Tian)
Combining: P x (y) =
∑
w,z,x ′ ,w ′ P(y|z,x ′ ,w ′ )P(x ′ ,w ′ )P(z|x,w)P(w)
W
Z
X
Y

Identification Failure
Identifying P x (y) fails if Y is a C-component,
Y ∪ X = V is a C-component, and some X is
relevant to Y (e.g., is in An(Y ))
Key theorem for completeness (aaai-06): if
the above failure occurs in any subproblem,
the overall effect is not identifiable
Proof by explicit counterexample (as in the
bow arc case)
Sanity check: the bow arc graph is a special
case of this condition

Conditional Effects
The likelihood of Y may be altered by
observing Z = z after doing do(x)
Such conditional effects are represented by
distributions of the form
P x (y|z) = P x (y,z)/P x (z)
In general, interventions and conditioning do
not commute: P(y|z,do(x)) ≠ P(y|do(x),z)
However, if Z ∉ De(X) G , then equality holds

Identifying Conditional Effects
Main result (uai-06): there exists a unique
maximum subset w ⊆ z, such that:
We can discover w in polynomial time
P x (y|z) = P x,w (y|z \ w)
P(v),G ⊢ id P x (y|z) iff P(v),G ⊢ id P x,w (y,z \ w)
This means identifiability of unconditional effects
is all we need, and we get completeness for free!

Identifying Counterfactuals
Counterfactuals involve multiple worlds
Our “axioms” φ will be the set of all
experiments we can perform in a single world:
P ∗ = {P x (v \ x)|x ⊆ v}
We separate two sets of difficulties:
Going from P(v) to P ∗ (already handled)
Going from P ∗ to P(γ) or P(γ|δ)

Graphs and Counterfactuals
We need a way to display causal
assumptions across multiple worlds
First attempt: twin network graph (Balke and
Pearl)
Problem: only two worlds!
A
A=false
A*=true
H H H*

Graphs and Counterfactuals (ct.)
Adding more worlds yields the parallel worlds
graph (ijcai-05)
More problems: distinct nodes can be the
same variable
Urx
Rx
Rx Rx’ Rx*
A
A=a
Uh
A’=a
A*=a*
H
H
H’ H*

Graphs and Counterfactuals (ct.)
Our solution: Rid the parallel worlds graph of
duplicate nodes inductively, starting at the
roots to obtain the counterfactual graph
Urx
Rx
Rx Rx’ Rx*
Rx
A
A=a
Uh
A’=a
A*=a*
A=a
Uh
A*=a*
H
H
H’ H*
H
H*

Identification of Counterfactuals
All identifiable P(γ) can be computed from P ∗
in a way similar to effects (subproblems based
on C-components in the counterfactual graph)
Testing P(γ|δ) similar to testing P x (y|z)
Testing P(γ) fails if γ forms a C-component,
all subscripts are in Pa(γ), and there is a
“conflict,” e.g., ∃y x ∈ γ and z x
′ ∈ γ or x ′ z ∈ γ
Key theorem (uai-07): failure in any
subproblem implies overall counterfactual not
identifiable

Example
Effect of treatment on the treated: P(y x |x ′ )
Not identifiable in the bow arc graph, but
identifiable in the front-door graph
P(y x |x ′ )= P(y x,x ′ )
P(x ′ )
=
∑
z P(y|z,x′ )P(x ′ )P(z|x)
P(x ′ )
∑
z P z(y,x ′ )P x (z)
P(x ′ )
=
= ∑ z P(y|z,x′ )P(z|x)
U
X=x’
U
X
Z Y
X=x Z Y

Conclusions
Using the framework of graphical causal
models, we posed two causal deduction
problems: evaluating causal effects from
observations, and evaluating counterfactuals
from experiments
We gave complete algorithms for solving
these problems, along with a characterization
of models where a given problem can be
solved