Talk at MIT-IBM seminar

1. Causality-inspired ML: what can
ideas from causality do for ML?
The domain adaptation case
Sara Magliacane
MIT-IBM Watson AI Lab

2. Causality-inspired ML
• Real-world ML needs to deal with:
• Biased data (fairness, selection bias, generalization)
• Heterogeneous data, small samples, missing/corrupted data, not iid
• Actionable insights (decisions cannot be made on correlations)

3. Causality-inspired ML
• Real-world ML needs to deal with:
• Biased data (fairness, selection bias, generalization)
• Heterogeneous data, small samples, missing/corrupted data, not iid
• Actionable insights (decisions cannot be made on correlations)
• Causal inference can help with some of these questions:
• Systematic data fusion and reuse with biased data, heterogenous, not iid data
• A systematic way to extract actionable insights

4. Causality-inspired ML
• Real-world ML needs to deal with:
• Biased data (fairness, selection bias, generalization)
• Heterogeneous data, small samples, missing/corrupted data, not iid
• Actionable insights (decisions cannot be made on correlations)
• Causal inference can help with some of these questions:
• Systematic data fusion and reuse with biased data, heterogenous, not iid data
• A systematic way to extract actionable insights
• “Full” causality can be not necessary or too expensive -> causality-inspired ML

5. Unsupervised domain adaptation
by inferring separating sets of
features
(joint work with Thijs van Ommen, Tom Claassen, Stephan Bongers, Philip Versteeg and Joris Mooij)

6. Transfer learning and causal inference
• Transfer learning:
• How can I predict what happens
when the distribution changes?

7. Transfer learning and causal inference
• Transfer learning:
• How can I predict what happens
when the distribution changes?

8. Transfer learning and causal inference
• Transfer learning:
• How can I predict what happens
when the distribution changes?
• Causal inference:
• How can I predict what happens when the
distribution changes after an intervention?
• Perfect intervention do(X):
• do-calculus [Pearl, 2009]

9. Transfer learning and causal inference
• Transfer learning:
• How can I predict what happens
when the distribution changes?
• Causal inference:
• How can I predict what happens when the
distribution changes after an intervention?
• Perfect intervention do(X):
• do-calculus [Pearl, 2009]
• Soft intervention on X change of
distribution of P(X| parents) in the
graph
• arbitrary change of distribution
≈
≈

10. ICML 2012

11. Domain adaptation
• Source domain (or domains):
• Target domain:
• Domain adaptation: predict Y in target domain using (also) information from source domain
• Unsupervised domain adaptation: no labels in the target domain (zero-shot)
{Xi, Yi}n
i=1 ∼ PS(X, Y)
{Xj, Yj}m
j=1 ∼ PT(X, Y) ≠ PS(X, Y)
{Xj}m
j=1 ∼ PT(X)

12. Domain adaptation
X1 X2 Y
Normal 0,1 2 0
Normal 0,2 3 0
Normal 1,1 2 1
Normal 0,1 3 0
X1 X2 Y
Gene A 3,1 2 ?
Gene A 3,2 3 ?
Gene A 4 2 ?
Gene A 3,2 3 ?
• Source domain (or domains):
• Target domain:
• Domain adaptation: predict Y in target domain using (also) information from source domain
• Unsupervised domain adaptation: no labels in the target domain (zero-shot)
{Xi, Yi}n
i=1 ∼ PS(X, Y)
{Xj, Yj}m
j=1 ∼ PT(X, Y) ≠ PS(X, Y)
{Xj}m
j=1 ∼ PT(X)

13. Domain adaptation from a graphical perspective
X1 X2 Y
Normal 0,1 2 0
Normal 0,2 3 0
Normal 1,1 2 1
Normal 0,1 3 0
X1 X2 Y
Gene A 3,1 2 ?
Gene A 3,2 3 ?
Gene A 4 2 ?
Gene A 3,2 3 ?

14. Domain adaptation from a graphical perspective
D X1 X2 Y
Normal 0,1 2 0
Normal 0,2 3 0
Normal 1,1 2 1
Normal 0,1 3 0
• Add a variable D to represent the domain
X1 X2 Y
Gene A 3,1 2 ?
Gene A 3,2 3 ?
Gene A 4 2 ?
Gene A 3,2 3 ?

15. Domain adaptation from a graphical perspective
• Add a variable D to represent the domain
• Consider the data as coming from a single distribution P(X,Y, D)
X1 X2 Y
Gene A 3,1 2 ?
Gene A 3,2 3 ?
Gene A 4 2 ?
Gene A 3,2 3 ?
D X1 X2 Y
Normal 0,1 2 0
Normal 0,2 3 0
Normal 1,1 2 1
Normal 0,1 3 0

16. Domain adaptation from a graphical perspective
• Add a variable D to represent the domain
• Consider the data as coming from a single distribution P(X,Y, D)
X1 Y X2D
• We can represent P(X,Y, D) with a
(possibly unknown) causal graph
X1 X2 Y
Gene A 3,1 2 ?
Gene A 3,2 3 ?
Gene A 4 2 ?
Gene A 3,2 3 ?
D X1 X2 Y
Normal 0,1 2 0
Normal 0,2 3 0
Normal 1,1 2 1
Normal 0,1 3 0

17. D-separation with slightly less tears
X1 YD
• Given a causal graph, d-separation is a graphical criterion that (under
standard conditions) allows us to read conditional independences
Y /⊥⊥ D

18. D-separation with slightly less tears
X1 YD
• Given a causal graph, d-separation is a graphical criterion that (under
standard conditions) allows us to read conditional independences
Y ⊥⊥ D|X1Y /⊥⊥ D
• Conditioning on X1 closes the path

19. D-separation with slightly less tears
X1 YD
• Given a causal graph, d-separation is a graphical criterion that (under
standard conditions) allows us to read conditional independences
Y ⊥⊥ D|X1Y /⊥⊥ D
• Conditioning on X1 closes the path
Y ⊥⊥ D
Y
X2
D

20. D-separation with slightly less tears
X1 YD
• Given a causal graph, d-separation is a graphical criterion that (under
standard conditions) allows us to read conditional independences
Y ⊥⊥ D|X1Y /⊥⊥ D
• Conditioning on X1 closes the path
Y ⊥⊥ D Y /⊥⊥ D|X2
• Conditioning on X2 opens a closed path
Y
X2
D

21. D-separation with slightly less tears
X1 YD
• Given a causal graph, d-separation is a graphical criterion that (under
standard conditions) allows us to read conditional independences
Y ⊥⊥ D|X1Y /⊥⊥ D
• Conditioning on X1 closes the path
Y
X2
D
Y ⊥⊥ D Y /⊥⊥ D|X2
• Conditioning on X2 opens a closed path
X1 YD X2

22. D-separation with slightly less tears
X1 YD
• Given a causal graph, d-separation is a graphical criterion that (under
standard conditions) allows us to read conditional independences
Y ⊥⊥ D|X1Y /⊥⊥ D
• Conditioning on X1 closes the path
Y
X2
D
Y ⊥⊥ D Y /⊥⊥ D|X2
• Conditioning on X2 opens a closed path
X1 YD X2
Y /⊥⊥ D|X1, X2

23. D-separation with slightly less tears
X1 YD
• Given a causal graph, d-separation is a graphical criterion that (under
standard conditions) allows us to read conditional independences
Y ⊥⊥ D|X1Y /⊥⊥ D
• Conditioning on X1 closes the path
Y
X2
D
Y ⊥⊥ D Y /⊥⊥ D|X2
• Conditioning on X2 opens a closed path
• Conditional independences with D tell us about invariance
Y ⊥⊥ D|X1 ⟹ P(Y|X1, D) = P(Y|X1)
⟹ P(Y|X1, D = d1) = P(Y|X1, D = d2)

24. Domain adaptation classification from a graphical
perspective [Zhang et al. 2013]
• Covariate shift assumption:
• Related to counterfactual inference [Johansson et al. 2016, 2018]
PS(Y|X) = PT(Y|X) ≡ Y ⊥⊥ D|X
X1 YX2D

25. Domain adaptation classification from a graphical
perspective [Zhang et al. 2013]
• Covariate shift assumption:
• Related to counterfactual inference [Johansson et al. 2016, 2018]
• Target shift and generalized target shift
PS(Y|X) = PT(Y|X) ≡ Y ⊥⊥ D|X
X1 YX2D
YD X YD X

26. Common misconceptions: 1. An invariant feature
need not be causal
X1 Y X2D
Y ⊥⊥ D|X1
• Y|X1,X2 is invariant invariant features are not necessarily parents of Y⟹
Y ⊥⊥ D|X1, X2

27. Common misconceptions: 1. An invariant feature
need not be causal
X1 Y X2D
Y ⊥⊥ D|X1
• Y|X1,X2 is invariant invariant features are not necessarily parents of Y⟹
Y ⊥⊥ D|X1, X2
Invariant feature across “many different datasets” is not enough in general to find
causal parents, need more assumptions

28. Common misconceptions: 1. An invariant feature
need not be causal
X1 Y X2D
Y ⊥⊥ D|X1
• Y|X1,X2 is invariant invariant features are not necessarily parents of Y
• Invariant Causal Prediction [Peters et al. 2016] under causal sufficiency:
⟹
S* =
⋂
Y⊥⊥D|S
S ⊆ Pa(Y) {X1, X2} ∩ {X1} = {X1}
Y ⊥⊥ D|X1, X2

29. Common misconceptions: 1. An invariant feature
need not be causal
X1 Y X2D
Y ⊥⊥ D|X1
• Y|X1,X2 is invariant invariant features are not necessarily parents of Y
• Invariant Causal Prediction [Peters et al. 2016] under causal sufficiency:
⟹
S* =
⋂
Y⊥⊥D|S
S ⊆ Pa(Y) {X1, X2} ∩ {X1} = {X1}
Y ⊥⊥ D|X1, X2

30. Common misconceptions: 1. An invariant feature
need not be causal
X1 Y X2D
Y ⊥⊥ D|X1
{X1, X2} ∩ {X1} = {X1}
Y ⊥⊥ D|X1, X2
X1 YX2D {X1} ∩ {X2} ∩ {X1, X2} = ∅
• Y|X1,X2 is invariant invariant features are not necessarily parents of Y
• Invariant Causal Prediction [Peters et al. 2016] under causal sufficiency:
⟹
S* =
⋂
Y⊥⊥D|S
S ⊆ Pa(Y)
Sometimes too
conservative:

31. Common misconception 2: Parents are not
enough under latent confounding
X1 YX2D
Y ⊥⊥ D|X1
Y /⊥⊥ D|X2
Y /⊥⊥ D|X1, X2
• Y|X1 is invariant, Y|X2 is not
Even if we knew all the parents, under latent confounding this wouldn’t
necessarily help transfer

32. Common misconception 2: Parents are not
enough under latent confounding
X1 YX2D
Y ⊥⊥ D|X1
Y /⊥⊥ D|X2
Y /⊥⊥ D|X1, X2
• Y|X1 is invariant, Y|X2 is not
Even if we knew all the parents, under latent confounding this wouldn’t
necessarily help transfer
• Conclusion: causality (e.g. using the causal parents, learning the
complete causal graph) is neither necessary or sufficient* for transfer

33. Desiderata for a causality inspired domain
adaptation method
• X, Y and changes can be represented by an unknown causal graph
• Allow for latent confounders
• Avoid parametric assumptions, allow for heterogeneous effects across domains

34. Desiderata for a causality inspired domain
adaptation method
• X, Y and changes can be represented by an unknown causal graph
• Allow for latent confounders
• Avoid parametric assumptions, allow for heterogeneous effects across domains
• Instead of modeling changes between each domain, distinguish the change
between the mixture of sources and the target

35. Desiderata for a causality inspired domain
adaptation method
• X, Y and changes can be represented by an unknown causal graph
• Allow for latent confounders
• Avoid parametric assumptions, allow for heterogeneous effects across domains
• Instead of modeling changes between each domain, distinguish the change
between the mixture of sources and the target
• Avoid common assumption that if Y| T(X) is invariant across multiple source
domains, then Y|T(X) is invariant also in the target domain
• This also (implicitly) assumed by methods based on the idea that
invariance => causality

36. Causal domain adaptation problem
[Magliacane et al. 2018]
X1 X2 Y
Normal 0,1 2 0
Normal 0,2 3 0
Normal 1,1 2 1
Normal 0,1 3 0
X1 X2 Y
Gene A 3,1 2 1
Gene A 3,2 3 1
Gene A 4 1 1
Gene A 3,2 3 0
X1 X2 Y
Gene B 0,2 1 ?
Gene B 0,3 1 ?
Gene B 0,3 2 ?
Gene B 0,4 1 ?
• Unsupervised multi-source domain adaptation
• We interpret the change in the target domain as a soft intervention
• We assume Y cannot be intervened upon directly - P(Y) can still change

37. C1 C2 X1 X2 Y
0 1 0,2 1 ?
0 1 0,3 1 ?
0 1 0,3 2 ?
0 1 0,4 1 ?
C1 C2 X1 X2 Y
1 0 3,1 2 1
1 0 3,2 3 1
1 0 4 1 1
1 0 3,2 3 0
Joint Causal Inference [Mooij et al. 2020]
• We represent jointly different distributions as an unknown single causal graph
• Instead of a single domain variable, we add several context variables so we can
disentangle changes in distribution across the datasets
• If we know nothing about the changes in the datasets, we use indicator variables
X1 X2 Y
Normal 0,1 2 0
Normal 0,2 3 0
Normal 1,1 2 1
Normal 0,1 3 0
X1 X2 Y
Gene A 3,1 2 1
Gene A 3,2 3 1
Gene A 4 1 1
Gene A 3,2 3 0X1 X2 Y
Gene B 0,2 1 ?
Gene B 0,3 1 ?
Gene B 0,3 2 ?
Gene B 0,4 1 ?
C1 C2 X1 X2 Y
0 0 0,1 2 0
0 0 0,2 3 0
0 0 1,1 2 1
0 0 0,1 3 0

38. C1=1C1=0
Joint Causal Inference [Mooij et al. 2020]
• The joint graph is the union of all the possibly different subgraphs in each
dataset, and context variables that model the differences in each dataset
• We allow for latent confounders and in general also cyclic causal graphs
X1 Y X2 X1 Y X2 X1 Y X2
C1

39. Joint Causal Inference [Mooij et al. 2020]
• We can learn an equivalence class of the unknown single causal graph using
conditional independence tests on systematically pooled dat
• We treat context variables as normal variables that we know are uncaused and can
use any standard constraint-based causal discovery method
X1 Y X2
C1C2C1 ⊥⊥ Y|X1
X1 ⊥⊥ X2 |Y, C2
X1 /⊥⊥ X2 |Y
…C1 C2 X1 X2 Y
0 1 0,2 0 0
0 1 0,3 0 1
0 1 0,3 1 0
C1 C2 X1 X2 Y
1 0 3,1 2 1
1 0 3,2 3 1
1 0 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1

40. X1 X2 Y
1 0,2 1 ?
1 0,3 1 ?
1 0,3 2 ?
1 0,4 1 ?
Causal domain adaptation: separating features
X1 Y X2
C1
• For simplicity: consider only one source and one target
• Assume we knew the causal graph:
C1 X1 X2 Y
0 0,1 2 0
0 0,2 3 0
0 1,1 2 1
0 0,1 3 0
• Which features can we use to predict Y in the target domain?

41. Causal domain adaptation: separating features
X1 Y X2
C1 Source
Target
X1
Y
Y ⊥⊥ C1 |X1 ≡
P(Y|X1, C1 = 0) = P(Y|X1, C1 = 1)
• Separating features: sets of features that d-separate Y from the context
variable C1 representing the target domain

42. Causal domain adaptation: separating features
X1 Y X2
C1 Source
Target
X1
Y
Y ⊥⊥ C1 |X1 ≡
P(Y|X1, C1 = 0) = P(Y|X1, C1 = 1)
Source
Target
Y /⊥⊥ C1 |X2 ≡
P(Y|X2, C1 = 0) ≠ P(Y|X2, C1 = 1)
X2
Y
• Separating features: sets of features that d-separate Y from the context
variable C1 representing the target domain

43. Causal domain adaptation: separating features
X1 Y X2
C1 Source
Target
X1
Y
P(Y|X1, C1 = 0) = P(Y|X1, C1 = 1)
Source
Target
P(Y|X2, C1 = 0) ≠ P(Y|X2, C1 = 1)
X2
Y
• Separating features: sets of features that d-separate Y from the context
variable C1 representing the target domain Features for which covariate
shift holds!
Y ⊥⊥ C1 |X1 ≡ Y /⊥⊥ C1 |X2 ≡

44. Causal domain adaptation: separating features
X1 Y X2
C1 Source
Target
X1
Y
P(Y|X1, C1 = 0) = P(Y|X1, C1 = 1)
Source
Target
P(Y|X2, C1 = 0) ≠ P(Y|X2, C1 = 1)
X2
Y
• Separating features: sets of features that d-separate Y from the context
variable C1 representing the target domain Features for which covariate
shift holds!
Y ⊥⊥ C1 |X1 ≡ Y /⊥⊥ C1 |X2 ≡
• {X1} is a separating feature set, {X1, X2} could lead to arbitrary large error

45. Causal domain adaptation: separating features
X1 Y X2
C1
• Separating features: sets of features that d-separate Y from the context
variable C1 representing the target domain
Source
Target
X1
Y
Source
Target
X2
Y
• {X1} is a separating feature set, {X1, X2} could lead to arbitrary large error
• Separating features need not be causal (parents of Y), e.g. X3
X3 Y ⊥⊥ C1 |X1 Y /⊥⊥ C1 |X2

46. What if the causal graph is unknown?
• Idea: we could test the conditional independence in the data
Y ⊥⊥ C1 |X1? Y ⊥⊥ C1 |X2?

47. What if the causal graph is unknown?
• Idea: we could test the conditional independence in the data
X1 X2 Y
1 0,2 1 ?
1 0,3 1 ?
1 0,3 2 ?
1 0,4 1 ?
C1 X1 X2 Y
0 0,1 2 0
0 0,2 3 0
0 1,1 2 1
0 0,1 3 0
Y ⊥⊥ C1 |X1? Y ⊥⊥ C1 |X2?
• Problem: Y is always missing when C1=1, so we cannot test these

48. What if the causal graph is unknown?
• Idea: we could test the conditional independence in the data
X1 X2 Y
1 0,2 1 ?
1 0,3 1 ?
1 0,3 2 ?
1 0,4 1 ?
C1 X1 X2 Y
0 0,1 2 0
0 0,2 3 0
0 1,1 2 1
0 0,1 3 0
Y ⊥⊥ C1 |X1? Y ⊥⊥ C1 |X2?
• Problem: Y is always missing when C1=1, so we cannot test these
X1 /⊥⊥ X2
X1 /⊥⊥ C1
X1 ⊥⊥ X2 |Y, C1 = 0
X1 /⊥⊥ X2 |C1
…
• Idea: Can we use all other independences?

49. Assumptions [Magliacane et al. 2018]
• We assume that there exists an acyclic causal graph that fits all the data
(Joint Causal Inference)
• We assume Y cannot be intervened upon directly

50. Assumptions [Magliacane et al. 2018]
A ⊥⊥ D|B, Y, C1 = 0 ⟹ A ⊥⊥ D|B, Y, C1 = 1
Y ⊥⊥ A|B, C1 = 0 ⟹ Y ⊥⊥ A|B, C1 = 1A, D, B ⊂ V∖{Y, C1}
• We assume that there exists an acyclic causal graph that fits all the data
(Joint Causal Inference)
• We assume Y cannot be intervened upon directly
• We assume no extra dependences involving Y in target domain C1=1

51. Assumptions [Magliacane et al. 2018]
• We assume that there exists an acyclic causal graph that fits all the data
(Joint Causal Inference)
• We assume Y cannot be intervened upon directly
• We assume no extra dependences involving Y in target domain C1=1
A ⊥⊥ D|B, Y, C1 = 0 ⟹ A ⊥⊥ D|B, Y, C1 = 1
Y ⊥⊥ A|B, C1 = 0 ⟹ Y ⊥⊥ A|B, C1 = 1A, D, B ⊂ V∖{Y, C1}
C1 ⊥⊥ Y|B?
• Note that this does not assume anything about the separating set test :

52. A slightly more complicated example
C1 C2 X1 X2 ?
1 0 0,2 0 ?
1 0 0,3 0 ?
1 0 0,3 1 ?
C1 C2 X1 X2 Y
0 1 3,1 2 1
0 1 3,2 3 1
0 1 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1
C1 C2 X1 X2 ?
1 0 0,2 0 ?
1 0 0,3 0 ?
1 0 0,3 1 ?
C1 C2 X1 X2 Y
0 1 3,1 2 1
0 1 3,2 3 1
0 1 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1

53. A slightly more complicated example
C1 C2 X1 X2 ?
1 0 0,2 0 ?
1 0 0,3 0 ?
1 0 0,3 1 ?
C1 C2 X1 X2 Y
0 1 3,1 2 1
0 1 3,2 3 1
0 1 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1
Y ⊥⊥ C2 |X1, C1 = 0
Y /⊥⊥ C2 |C1 = 0
X2 ⊥⊥ C2 |Y, C1 = 0C1 C2 X1 X2 ?
1 0 0,2 0 ?
1 0 0,3 0 ?
1 0 0,3 1 ?
C1 C2 X1 X2 Y
0 1 3,1 2 1
0 1 3,2 3 1
0 1 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1
Perform allowed CI tests

54. A slightly more complicated example
X1 Y X2
C1C2
C1 C2 X1 X2 ?
1 0 0,2 0 ?
1 0 0,3 0 ?
1 0 0,3 1 ?
C1 C2 X1 X2 Y
0 1 3,1 2 1
0 1 3,2 3 1
0 1 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1
Y ⊥⊥ C2 |X1, C1 = 0
Y /⊥⊥ C2 |C1 = 0
X2 ⊥⊥ C2 |Y, C1 = 0C1 C2 X1 X2 ?
1 0 0,2 0 ?
1 0 0,3 0 ?
1 0 0,3 1 ?
C1 C2 X1 X2 Y
0 1 3,1 2 1
0 1 3,2 3 1
0 1 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1
Perform allowed CI tests All possible compatible graphs

55. A slightly more complicated example
X1 Y X2
C1C2
C1 C2 X1 X2 ?
1 0 0,2 0 ?
1 0 0,3 0 ?
1 0 0,3 1 ?
C1 C2 X1 X2 Y
0 1 3,1 2 1
0 1 3,2 3 1
0 1 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1
Y ⊥⊥ C2 |X1, C1 = 0
Y /⊥⊥ C2 |C1 = 0
X2 ⊥⊥ C2 |Y, C1 = 0
Y ⊥⊥ C1 |X1
• We can prove untestable separating test without reconstructing the graph:
C1 C2 X1 X2 ?
1 0 0,2 0 ?
1 0 0,3 0 ?
1 0 0,3 1 ?
C1 C2 X1 X2 Y
0 1 3,1 2 1
0 1 3,2 3 1
0 1 4 3 1
C1 C2 X1 X2 Y
0 0 0,1 1 0
0 0 0,2 1 0
0 0 1,1 2 1
Perform allowed CI tests All possible compatible graphs
True in all possible compatible graphs

56. C1=0
Invariance in sources is not invariance in target
X1 Y X2
C1C2
Y ⊥⊥ C2 |X1, C1 = 0
Y /⊥⊥ C2 |C1 = 0
X2 ⊥⊥ C2 |Y, C1 = 0
Y ⊥⊥ C2 |X1, C1 = 0
• Features that are invariant across the source domains:
Y ⊥⊥ C2 |X2, C1 = 0 Y ⊥⊥ C2 |X1, X2, C1 = 0
X1 Y X2
C2

57. C1=0
Invariance in sources is not invariance in target
X1 Y X2
C1C2
Y ⊥⊥ C2 |X1, C1 = 0
Y /⊥⊥ C2 |C1 = 0
X2 ⊥⊥ C2 |Y, C1 = 0
Y ⊥⊥ C2 |X1, C1 = 0
• Features that are invariant across the source domains:
Y ⊥⊥ C2 |X2, C1 = 0 Y ⊥⊥ C2 |X1, X2, C1 = 0
X1 Y X2
C2
Y ⊥⊥ C2 |X1, C1 = 1 Y /⊥⊥ C2 |X2, C1 = 1 Y /⊥⊥ C2 |X1, X2, C1 = 1

58. C1=0
Invariance in sources is not invariance in target
X1 Y X2
C1C2
Y ⊥⊥ C2 |X1, C1 = 0
Y /⊥⊥ C2 |C1 = 0
X2 ⊥⊥ C2 |Y, C1 = 0
Y ⊥⊥ C2 |X1, C1 = 0
• Features that are invariant across the source domains:
Y ⊥⊥ C2 |X2, C1 = 0 Y ⊥⊥ C2 |X1, X2, C1 = 0
X1 Y X2
C2
Y ⊥⊥ C2 |X1, C1 = 1 Y /⊥⊥ C2 |X2, C1 = 1 Y /⊥⊥ C2 |X1, X2, C1 = 1
• Idea: use data from target also in order to learn invariant features

59. Inferring separating sets without enumerating all
possible causal graphs
Query Y ⊥⊥ C1 |X1?
Y ⊥⊥ C2 |X1, C1 = 0
X1 ⊥⊥ X3 |X4
X2 ⊥⊥ C2 |Y, C1 = 0
…

60. Inferring separating sets without enumerating all
possible causal graphs
Query Y ⊥⊥ C1 |X1?
All testable conditional
independences from data
Y ⊥⊥ C2 |X1, C1 = 0
X1 ⊥⊥ X3 |X4
X2 ⊥⊥ C2 |Y, C1 = 0
…
Assumptions

61. Logic encoding of d-separation
[Hyttinen et al. 2014]
Inferring separating sets without enumerating all
possible causal graphs
Query Y ⊥⊥ C1 |X1?
All testable conditional
independences from data
Y ⊥⊥ C2 |X1, C1 = 0
X1 ⊥⊥ X3 |X4
X2 ⊥⊥ C2 |Y, C1 = 0
…
Assumptions
Theorem prover

62. Logic encoding of d-separation
[Hyttinen et al. 2014]
Inferring separating sets without enumerating all
possible causal graphs
Query Y ⊥⊥ C1 |X1?
All testable conditional
independences from data
Y ⊥⊥ C2 |X1, C1 = 0
X1 ⊥⊥ X3 |X4
X2 ⊥⊥ C2 |Y, C1 = 0
…
Assumptions
?
Y ⊥⊥ C1 |X1
Y /⊥⊥ C1 |X1
Theorem prover
Not identifiable
Provably separating
Provably not separating

63. Causal feature selection
• Use standard feature selection method to create a list L of subsets of
features and context variables ordered by lowest source domain loss
• For example, L = {X1, C2}, {X1, X2, C2} , {X1, X2}, …
• Iteratively query theorem prover until provably separating set A
• Train a model f(A) to predict Y on source domains
• Apply f(A) to predict Y on target domain

64. Causal feature selection
• Use standard feature selection method to create a list L of subsets of
features and context variables ordered by lowest source domain loss
• For example, L = {X1, C2}, {X1, X2, C2} , {X1, X2}, …
• Iteratively query theorem prover until provably separating set A
• Train a model f(A) to predict Y on source domains
• Apply f(A) to predict Y on target domain

65. Results
• Simulated data: 200 random causal graphs
• Baseline method for feature selection and regression: random forest
• Real-world data: CRM Causal Inference Challenge
• 441 wild-type mice with 16 phenotypes
• ~13 mutant mice (single-gene knockout) for each of 10 interventions
• “Causal cross-validation”
• We randomly selected 3 phenotypes and 2 knockouts and created 1000 datasets
• For each dataset we randomly chose Y and C1 and ignored the values of Y in C1=1

66. Desiderata for a causality inspired domain
adaptation method
• X, Y and changes can be represented by an unknown causal graph
• Allow for latent confounders
• Avoid parametric assumptions, allow for heterogeneous effects across domains
• Instead of modeling changes between each domain, distinguish the change
between the mixture of sources and the target
• Avoid common assumption that if Y| T(X) is invariant across multiple source
domains, then Y|T(X) is invariant also in the target domain
• Only search for invariant features with respect to current target task
No need to find
causal graph or
equivalence
class

67. Limitations and future work
• Potentially too conservative: Separating sets may exist that are not provably separating
• Extension: can we use active learning/intervention design to decide most
informative interventions?
• Cannot use directly for transformations of features T(X)
• Extension: incorporate structure learning with deterministic relations
• Scalability: using (error-correcting) logic-based encoding with all CI tests as input scales to
tens of vars (including context variables)
• Extension: use approximate algorithms, combine with low dim representations
• Can we apply this to multi-task RL (e.g. in factored MDPs)?

68. Thanks!