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 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 Weingärtner, et al (2010). Relationship between cholesterol synthesis and intestinal absorption isassociated with cardiovascular risk. Atherosclerosis . Sebastian Weichwald — Causal Models under Variable Transformations — Slide 2

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 Weingärtner, et al (2010). Relationship between cholesterol synthesis and intestinal absorption isassociated with cardiovascular risk. Atherosclerosis . Sebastian Weichwald — Causal Models under Variable Transformations — Slide 3

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 � − LDL diet heart disease HDL + Rubenstein*, Weichwald*, et al (2017). Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence . Sebastian Weichwald — Causal Models under Variable Transformations — Slide 4

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 + − LDL diet heart disease HDL + Rubenstein*, Weichwald*, et al (2017). Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence . Sebastian Weichwald — Causal Models under Variable Transformations — Slide 4

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 + ⇑ − LDL diet heart disease HDL + Rubenstein*, Weichwald*, et al (2017). Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence . Sebastian Weichwald — Causal Models under Variable Transformations — Slide 4

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 � C 2 C i causal entities C 1 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 � linear mixing C 2 C i causal entities C 1 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 � F 1 F 2 F 3 observed linear mixture linear mixing C 2 C i causal entities C 1 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 � observed rgb pixel values F 1 F 2 F 3 taking a photo C 2 C i causal entities C 1 C 4 C 5 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 Runge et al. (2019). Inferring causation from time series in Earth system sciences. Nature Communications. Sebastian Weichwald — Causal Models under Variable Transformations — Slide 6

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

M X M Y 휏 ? X 6 X 1 X 4 휏 1 ( X ) 휏 2 ( X ) X 3 X 5 X 2 휏 3 ( 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 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 Rubenstein*, Weichwald*, et al (2017). Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence . Sebastian Weichwald — Causal Models under Variable Transformations — Slide 8

Causal Models as Posets of Distributions

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: M X : 휃 ↦→ P 휃 P 휃 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 Causal Model: “Normal” Probabilistic Model: M X : 휃 ↦→ { P do ( i ) M X : 휃 ↦→ P 휃 : i ∈ I X } 휃 I X is set of interventions. P do ( i 3 ) 휃 P do ( i 2 ) 휃 P ∅ P 휃 휃 P do ( i 1 ) 휃 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 Causal Models P do ( C = 0 ) X P do ( A = 0 , C = 0 ) X P ∅ X P do ( A = 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 do ( C = 0 ) X P do ( A = 0 , C = 0 ) X P ∅ X P do ( A = 0 ) X X has partial ordering structure I 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 do ( C = 0 ) X P do ( A = 0 , C = 0 ) X P ∅ X P do ( A = 0 ) X X has partial ordering structure I �� � � P do ( i ) M X implies the poset of distributions P X : = : i ∈ I , ≤ X X 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 Structural Causal Models M X = (S X , I X , P E X ) 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     X 1 = E 1 • S X =   X 2 = X 1 + E 2 M X = (S X , I X , P E X )  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     X 1 = E 1 • S X =   X 2 = X 1 + E 2 M X = (S X , I X , P E X )  • I X = {∅ , do ( X 1 = 5 ) , do ( X 2 = 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     X 1 = E 1 • S X =   X 2 = X 1 + E 2 M X = (S X , I X , P E X )  • I X = {∅ , do ( X 1 = 5 ) , do ( X 2 = 3 )} • E ∼ N ( 0 , I ) 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     X 1 = E 1 • S X =   X 2 = X 1 + E 2 M X = (S X , I X , P E X )  • I X = {∅ , do ( X 1 = 5 ) , do ( X 2 = 3 )} • E ∼ N ( 0 , I ) observational P ∅ X 1 ∼ N ( 0 , 1 ) P ∅ X 2 ∼ 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     X 1 = E 1 • S X =   X 2 = X 1 + E 2 M X = (S X , I X , P E X )  • I X = {∅ , do ( X 1 = 5 ) , do ( X 2 = 3 )} • E ∼ N ( 0 , I ) observational intervention on X 1 P do ( X 1 = 5 ) P ∅ X 1 ∼ N ( 0 , 1 ) ≡ 5 X 1 P ∅ X 2 ∼ N ( 0 , 2 ) P do ( X 1 = 5 ) ∼ N ( 5 , 1 ) X 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     X 1 = E 1 • S X =   X 2 = X 1 + E 2 M X = (S X , I X , P E X )  • I X = {∅ , do ( X 1 = 5 ) , do ( X 2 = 3 )} • E ∼ N ( 0 , I ) observational intervention on X 1 intervention on X 2 P do ( X 1 = 5 ) P do ( X 2 = 3 ) P ∅ X 1 ∼ N ( 0 , 1 ) ≡ 5 ∼ N ( 0 , 1 ) X 1 X 1 P ∅ X 2 ∼ N ( 0 , 2 ) P do ( X 1 = 5 ) P do ( X 2 = 3 ) ∼ N ( 5 , 1 ) ≡ 3 X 2 X 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 causal discovery? observations causal model P do ( i 2 ) X P do ( i 3 ) X P ∅ P ∅ X X P do ( i 1 ) P do ( i 1 ) X X � � � � P do ( i ) P do ( i ) : i ∈ I sub � I : i ∈ I X � I sub X X X Sebastian Weichwald — Causal Models under Variable Transformations — Slide 12