Causal Models under Variable Transformations Challenges for Causally - - PowerPoint PPT Presentation
c o p e n h a g e n c a u s a l i t y l a b university of copenhagen Causal Models under Variable Transformations Challenges for Causally Consistent Representation Learning Sebastian Weichwald sweichwald.de @sweichwald ETH Guest Lecture
c o p e n h a g e n c a u s a l i t y l a b
university of copenhagen
Challenges for Causally Consistent Representation Learning Sebastian Weichwald
sweichwald.de
@sweichwald
ETH Guest Lecture 2020-10-06
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 2 Weingärtner, et al (2010). Relationship between cholesterol synthesis and intestinal absorption isassociated with cardiovascular risk. Atherosclerosis.
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 3 Weingärtner, et al (2010). Relationship between cholesterol synthesis and intestinal absorption isassociated with cardiovascular risk. Atherosclerosis.
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Variable Transformations may break Causal Reasoning diet LDL HDL heart disease − +
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 4 Rubenstein*, Weichwald*, et al (2017). Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence.
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Variable Transformations may break Causal Reasoning diet total chol. heart disease − + diet LDL HDL heart disease − +
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 4 Rubenstein*, Weichwald*, et al (2017). Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence.
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Variable Transformations may break Causal Reasoning diet total chol. heart disease − +
diet LDL HDL heart disease − +
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 4 Rubenstein*, Weichwald*, et al (2017). Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence.
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Observables may not be (meaningful) Causal Entities
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 5
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Observables may not be (meaningful) Causal Entities
C1 C2 Ci
causal entities
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 5
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Observables may not be (meaningful) Causal Entities
C1 C2 Ci
causal entities linear mixing
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 5
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Observables may not be (meaningful) Causal Entities
C1 C2 Ci
F1 F2 F3 causal entities linear mixing
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 5
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Observables may not be (meaningful) Causal Entities
C1 C2 Ci
F1 F2 F3 causal entities taking a photo
C4 C5
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 5
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 6 Runge et al. (2019). Inferring causation from time series in Earth system sciences. Nature Communications.
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Variable Transformations may link Causal Reasoning at Different Scales fine-grained coarse-grained
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 7
MX MY
X1 X2 X3 X4 X5 X6 휏1(X) 휏2(X) 휏3(X) 휏 ?
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Consistency of Structural Equation Models auai.org/uai2017/proceedings/papers/11.pdf Paul Rubenstein, S Weichwald, S Bongers, JM Mooij, D Janzing, M Grosse-Wentrup, B Schölkopf
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 8 Rubenstein*, Weichwald*, et al (2017). Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence.
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
“Normal” Probabilistic Model: MX : 휃 ↦→ P휃
P휃
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 9
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
“Normal” Probabilistic Model: MX : 휃 ↦→ P휃
P휃
Causal Model: MX : 휃 ↦→ {Pdo(i)
휃
: i ∈ I
X}
I
X is set of interventions. P∅
휃
Pdo(i1)
휃
Pdo(i2)
휃
Pdo(i3)
휃
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 9
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Causal Models
P∅
X
Pdo(A=0)
X
Pdo(A=0,C=0)
X
Pdo(C=0)
X
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 10
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Models
P∅
X
Pdo(A=0)
X
Pdo(A=0,C=0)
X
Pdo(C=0)
X
I
X has partial ordering structure
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 10
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Models
P∅
X
Pdo(A=0)
X
Pdo(A=0,C=0)
X
Pdo(C=0)
X
I
X has partial ordering structure
MX implies the poset of distributions PX :=
X
: i ∈ I
X
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Structural Causal Models MX = (SX, I
X, PEX)
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 11
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Structural Causal Models MX = (SX, I
X, PEX)
X1 = E1 X2 = X1 + E2
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 11
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Structural Causal Models MX = (SX, I
X, PEX)
X1 = E1 X2 = X1 + E2
X = {∅, do(X1 = 5), do(X2 = 3)}
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 11
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Structural Causal Models MX = (SX, I
X, PEX)
X1 = E1 X2 = X1 + E2
X = {∅, do(X1 = 5), do(X2 = 3)}
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 11
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Structural Causal Models MX = (SX, I
X, PEX)
X1 = E1 X2 = X1 + E2
X = {∅, do(X1 = 5), do(X2 = 3)}
P∅
X1 ∼ N (0, 1)
P∅
X2 ∼ N (0, 2)
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 11
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Structural Causal Models MX = (SX, I
X, PEX)
X1 = E1 X2 = X1 + E2
X = {∅, do(X1 = 5), do(X2 = 3)}
P∅
X1 ∼ N (0, 1)
P∅
X2 ∼ N (0, 2)
intervention on X1 Pdo(X1=5)
X1
≡ 5 Pdo(X1=5)
X2
∼ N (5, 1)
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 11
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Structural Causal Models MX = (SX, I
X, PEX)
X1 = E1 X2 = X1 + E2
X = {∅, do(X1 = 5), do(X2 = 3)}
P∅
X1 ∼ N (0, 1)
P∅
X2 ∼ N (0, 2)
intervention on X1 Pdo(X1=5)
X1
≡ 5 Pdo(X1=5)
X2
∼ N (5, 1) intervention on X2 Pdo(X2=3)
X1
∼ N (0, 1) Pdo(X2=3)
X2
≡ 3
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 11
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
causal model causal discovery? P∅
X
Pdo(i1)
X
X
: i ∈ I
sub I X
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
X
: i ∈ I
X I sub
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
A B C B A C h A B C P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
A = 1/
2B − 1/ 2C+
2NA
B = √ 3NB C = 1/
3B
+
3NC
A = √ 2NA B = 1/
2A + C+
2NB
C = NC A = h +NA B = h + C+NB C = NC
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 13
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
A B C B A C h A B C P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
A = 1/
2B − 1/ 2C+
2NA
B = √ 3NB C = 1/
3B
+
3NC
A = √ 2NA B = 1/
2A + C+
2NB
C = NC A = h +NA B = h + C+NB C = NC
3 models inducing the same observational yet different interventional distributions
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 13
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
A B C B A C h A B C P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
P∅
X
Pdo(i1)
X
Pdo(i3)
X
Pdo(i2)
X
A = 1/
2B − 1/ 2C+
2NA
B = √ 3NB C = 1/
3B
+
3NC
A = √ 2NA B = 1/
2A + C+
2NB
C = NC A = h +NA B = h + C+NB C = NC
3 models inducing the same observational yet different interventional distributions fiting observational data well is not enough
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 13
MX MY
X1 X2 X3 X4 X5 X6 휏1(X) 휏2(X) 휏3(X) 휏 ?
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 14 Pearl (2009). Causality: Models, Reasoning, and Inference.
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The Three Layer Causal Hierarchy
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 15 Pearl (2018). Theoretical Impediments to Machine Learning – With Seven Sparks from the Causal Revolution. WSDM 2018.
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The Three Layer Causal Hierarchy presupposes that X and Y are the right variables
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 15 Pearl (2018). Theoretical Impediments to Machine Learning – With Seven Sparks from the Causal Revolution. WSDM 2018.
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Needed: Understanding of SCMs under Variable Transformations
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 16
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Needed: Understanding of SCMs under Variable Transformations Suppose we are given MX and a transformation 휏 : X → Y
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 16
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Needed: Understanding of SCMs under Variable Transformations Suppose we are given MX and a transformation 휏 : X → Y X ∼ PX an r.v. in X =⇒ 휏(X) ∼ P휏(X) is an r.v. in Y
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 16
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Needed: Understanding of SCMs under Variable Transformations Suppose we are given MX and a transformation 휏 : X → Y X ∼ PX an r.v. in X =⇒ 휏(X) ∼ P휏(X) is an r.v. in Y 휏 : PX → P휏(X) =
휏(X)
: i ∈ I
X
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Needed: Understanding of SCMs under Variable Transformations Suppose we are given MX and a transformation 휏 : X → Y X ∼ PX an r.v. in X =⇒ 휏(X) ∼ P휏(X) is an r.v. in Y 휏 : PX → P휏(X) =
휏(X)
: i ∈ I
X
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 16
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Needed: Understanding of SCMs under Variable Transformations Suppose we are given MX and a transformation 휏 : X → Y X ∼ PX an r.v. in X =⇒ 휏(X) ∼ P휏(X) is an r.v. in Y 휏 : PX → P휏(X) =
휏(X)
: i ∈ I
X
Does there exist an SCM MY with PY = P휏(X)? If so, then MY will agree with our observations of MX via 휏.
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 16
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What can go wrong? MX : X1 X2 X3 SX = X1 = E1 X2 = E2 X3 = X1 + X2 + E3 E2 = 1; E1, E3 arbitrary I
X =
do(∅) do(X1 = 0) do(X1 = 0, X2 = 0)
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 SX = X1 = E1 X2 = E2 X3 = X1 + X2 + E3 E2 = 1; E1, E3 arbitrary I
X =
do(∅) do(X1 = 0) do(X1 = 0, X2 = 0) MY : Y1 = X1 + X2 Y2 = X3
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 SX = X1 = E1 X2 = E2 X3 = X1 + X2 + E3 E2 = 1; E1, E3 arbitrary I
X =
do(∅) do(X1 = 0) do(X1 = 0, X2 = 0) MY : Y1 = X1 + X2 Y2 = X3 SY =
Y2 = Y1 + E3
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 SX = X1 = E1 X2 = E2 X3 = X1 + X2 + E3 E2 = 1; E1, E3 arbitrary I
X =
do(∅) do(X1 = 0) do(X1 = 0, X2 = 0) MY : Y1 = X1 + X2 Y2 = X3 SY =
Y2 = Y1 + E3 E1, E2, E3 as before
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 SX = X1 = E1 X2 = E2 X3 = X1 + X2 + E3 E2 = 1; E1, E3 arbitrary I
X =
do(∅) do(X1 = 0) do(X1 = 0, X2 = 0) MY : Y1 = X1 + X2 Y2 = X3 SY =
Y2 = Y1 + E3 E1, E2, E3 as before I
Y =
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 SX = X1 = E1 X2 = E2 X3 = X1 + X2 + E3 E2 = 1; E1, E3 arbitrary I
X =
do(∅) do(X1 = 0) do(X1 = 0, X2 = 0) MY : Y1 = X1 + X2 Y2 = X3 SY =
Y2 = Y1 + E3 E1, E2, E3 as before I
Y =
do(∅)
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 SX = X1 = E1 X2 = E2 X3 = X1 + X2 + E3 E2 = 1; E1, E3 arbitrary I
X =
do(∅) do(X1 = 0) do(X1 = 0, X2 = 0) MY : Y1 = X1 + X2 Y2 = X3 SY =
Y2 = Y1 + E3 E1, E2, E3 as before I
Y =
do(∅) do(Y1 = 1)
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 SX = X1 = E1 X2 = E2 X3 = X1 + X2 + E3 E2 = 1; E1, E3 arbitrary I
X =
do(∅) do(X1 = 0) do(X1 = 0, X2 = 0) MY : Y1 = X1 + X2 Y2 = X3 SY =
Y2 = Y1 + E3 E1, E2, E3 as before I
Y =
do(∅) do(Y1 = 1) do(Y1 = 0)
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 MY : Y1 = X1 + X2 Y2 = X3
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 MY : Y1 = X1 + X2 Y2 = X3 PX Pdo(X1=0)
X
Pdo(X1=0,X2=0)
X
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 MY : Y1 = X1 + X2 Y2 = X3 PX Pdo(X1=0)
X
Pdo(X1=0,X2=0)
X
PY 휏
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 MY : Y1 = X1 + X2 Y2 = X3 PX Pdo(X1=0)
X
Pdo(X1=0,X2=0)
X
PY 휏 Pdo(Y1=1)
Y
휏
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
What can go wrong? MX : X1 X2 X3 MY : Y1 = X1 + X2 Y2 = X3 PX Pdo(X1=0)
X
Pdo(X1=0,X2=0)
X
PY 휏 Pdo(Y1=1)
Y
휏 Pdo(Y1=0)
Y
휏
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 MY : Y1 = X1 + X2 Y2 = X3 PX Pdo(X1=0)
X
Pdo(X1=0,X2=0)
X
PY 휏 Pdo(Y1=1)
Y
휏 Pdo(Y1=0)
Y
휏
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
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What can go wrong? MX : X1 X2 X3 MY : Y1 = X1 + X2 Y2 = X3 PX Pdo(X1=0)
X
Pdo(X1=0,X2=0)
X
PY 휏 Pdo(Y1=1)
Y
휏 Pdo(Y1=0)
Y
휏
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
What can go wrong? MX : X1 X2 X3 MY : Y1 = X1 + X2 Y2 = X3 PX Pdo(X1=0)
X
Pdo(X1=0,X2=0)
X
PY 휏 Pdo(Y1=1)
Y
휏 Pdo(Y1=0)
Y
휏
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 17
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
1 structural equations SX;
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
1 structural equations SX; 2 restricted partially ordered set (I
X, ≤X) of interventions;
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
1 structural equations SX; 2 restricted partially ordered set (I
X, ≤X) of interventions;
3 exogenous variables distributed according to PEX.
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
1 structural equations SX; 2 restricted partially ordered set (I
X, ≤X) of interventions;
3 exogenous variables distributed according to PEX.
Let MY = SY, I
Y, PEY
be another SCM and 휏 : X → Y.
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
1 structural equations SX; 2 restricted partially ordered set (I
X, ≤X) of interventions;
3 exogenous variables distributed according to PEX.
Let MY = SY, I
Y, PEY
be another SCM and 휏 : X → Y. MY is an exact 휏-transformation of MX if
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
1 structural equations SX; 2 restricted partially ordered set (I
X, ≤X) of interventions;
3 exogenous variables distributed according to PEX.
Let MY = SY, I
Y, PEY
be another SCM and 휏 : X → Y. MY is an exact 휏-transformation of MX if Pi
휏(X) = Pdo(휔(i)) Y
∀i ∈ I
X
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
1 structural equations SX; 2 restricted partially ordered set (I
X, ≤X) of interventions;
3 exogenous variables distributed according to PEX.
Let MY = SY, I
Y, PEY
be another SCM and 휏 : X → Y. MY is an exact 휏-transformation of MX if Pi
휏(X) = Pdo(휔(i)) Y
∀i ∈ I
X
for a surjective order-preserving map 휔 : I
X → I Y.
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Exact Transformations ensure Causally Consistent SCMs Let MX = SX, I
X, PEX
be an SCM over variables X = (Xi : i ∈ IX) with
1 structural equations SX; 2 restricted partially ordered set (I
X, ≤X) of interventions;
3 exogenous variables distributed according to PEX.
Let MY = SY, I
Y, PEY
be another SCM and 휏 : X → Y. MY is an exact 휏-transformation of MX if Pi
휏(X) = Pdo(휔(i)) Y
∀i ∈ I
X
for a surjective order-preserving map 휔 : I
X → I Y.
=⇒ MX and MY are causally consistent
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 18
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Consistency PX Pdo(i)
X
Pdo(j)
X
PY Pdo(휔(i))
Y
Pdo(휔(j))
Y
do(i) do(j) do(휔(i)) do(휔(j)) 휏 휏 휏
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 19
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Elementary Properties of Exact Transformations
The identity mapping is an exact transformation.
MX MX id
Exact transformations are transitively closed.
MX MY MZ 휏1 휏2 휏2 ◦ 휏1
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 20
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Few Transformations yield Causally Consistent Representations
X1 X2 X3 MX
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 21
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Few Transformations yield Causally Consistent Representations
X1 X2 X3 MX subsystem MY
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 21
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Few Transformations yield Causally Consistent Representations
MX:
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 21
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Few Transformations yield Causally Consistent Representations
MX:
MY:
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 21
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Few Transformations yield Causally Consistent Representations
. . . . . . . . . . . . . . . . . . . . . . . . do(i) dynamic MX X1
t
X2
t
X1
t
X2
t
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 21
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Few Transformations yield Causally Consistent Representations
. . . . . . . . . . . . . . . . . . . . . . . . do(i) dynamic MX stationary MY Y1 Y2 Y1 Y2 do(휔(i))
휏 휏
X1
t
X2
t
X1
t
X2
t
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 21
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Models under Variable Transformations Challenges for Causally Consistent Representation Learning Lab Causality Copenhagen @sweichwald
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 22
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Models under Variable Transformations Challenges for Causally Consistent Representation Learning
Lab Causality Copenhagen @sweichwald
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 22
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Models under Variable Transformations Challenges for Causally Consistent Representation Learning
Lab Causality Copenhagen @sweichwald
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 22
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Models under Variable Transformations Challenges for Causally Consistent Representation Learning
Lab Causality Copenhagen @sweichwald
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 22
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Models under Variable Transformations Challenges for Causally Consistent Representation Learning
Lab Causality Copenhagen @sweichwald
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 22
u n i v e r s i t y o f c o p e n h a g e n c o p e n h a g e n c a u s a l i t y l a b
Causal Models under Variable Transformations Challenges for Causally Consistent Representation Learning
Lab Causality Copenhagen @sweichwald
Sebastian Weichwald — Causal Models under Variable Transformations — Slide 22