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Causal Programming Causal Programming Joshua Brul Joshua Brul - PowerPoint PPT Presentation

Sep 03, 2023 •46 likes •716 views

5/2/2019 talk slides Causal Programming Causal Programming Joshua Brul Joshua Brul file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 1/67 5/2/2019 talk slides Smoking/cancer structural causal model

  1. 5/2/2019 talk slides Causal Programming Causal Programming Joshua Brulé Joshua Brulé file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 1/67

  2. 5/2/2019 talk slides Smoking/cancer structural causal model Smoking/cancer structural causal model smoking = f 1 ϵ 1 ( ) tar = f 2 (smoking, ϵ 2 ) cancer = f 3 (tar, ϵ 3 ) ϵ 1 ⊥̸ ⊥ ϵ 3 tar smoking cancer file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 2/67

  3. 5/2/2019 talk slides Causal calculus (Pearl 1995) Causal calculus (Pearl 1995) P ( y ∣ x ^ , z , w ) = P ( y ∣ x ^ , w ) if ( Y ⊥ ⊥ Z ∣ X , W ) G X ¯¯ ¯ P ( y ∣ x ^ z , , w ) = P ( y ∣ ^ x ^ , z , w ) if ( Y ⊥ ⊥ Z ∣ X , W ) G X ¯¯ ¯ Z − − P ( y ∣ x ^ , z , w ) = P ( y ∣ x ^ , w ) if ( Y ⊥ ⊥ Z ∣ X , W ) G ¯¯ ¯ Z ( W ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ X , - nodes in a causal DAG W , X , Y , Z G delete edges pointing into G X ¯ ¯ ¯ ¯ ¯ X denotes delete edges emanating from G X X -nodes that are not ancestors of any -node − − Z ( W ) Z W Note: abbreviated ^ P(y ∣ do(x)) P(y ∣ x ) file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 3/67

  4. 5/2/2019 talk slides Example proof Example proof file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 4/67

  5. 5/2/2019 talk slides Causation coe�cient Causation coe�cient file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 5/67

  6. 5/2/2019 talk slides Correlation is not causation Correlation is not causation "Correlation is not causation but it sure is a hint." "Empirically observed covariation is a necessary but not suf�cient condition for causality." —Edward Tufte file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 6/67

  7. 5/2/2019 talk slides Correlation coe�cient Correlation coe�cient cov ( X , Y ) ρ = − − − − − − − − − − − − √ V ar [ X ] V ar [ Y ] ∑ x ∑ y xyP ( x , y ) − ∑ x xP ( x ) ∑ y yP ( y ) ρ = − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ∑ x x 2 ) 2 ∑ y y 2 ) 2 ( P ( x ) − ( ∑ x xP ( x ) )( P ( y ) − ( ∑ y yP ( y ) ) √ file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 7/67

  8. 5/2/2019 talk slides Correlation coe�cient (rewritten) Correlation coe�cient (rewritten) x 2 ) 2 V ar [ X ] = ∑ P ( x ) − ( ∑ xP ( x ) x x y 2 ) 2 V ar [ Y ] = ∑ ∑ P ( y | x ) P ( x ) − ( ∑ ∑ yP ( y | x ) P ( x ) x y x y ∑ x ∑ y xyP ( y | x ) P ( x ) − ∑ x xP ( x ) ∑ x ∑ y yP ( y | x ) P ( x ) ρ = − − − − − − − − − − − − √ V ar [ X ] V ar [ Y ] file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 8/67

  9. 5/2/2019 talk slides De�ning the causation coe�cient De�ning the causation coe�cient Substitute , abbreviated for P ( y ∣ do ( x )) P(y ∣ ^ x ) P ( y ∣ x ) i.e. Replace observational distribution with interventional distribution Substitute for ^ P ( x ) P ( x ) 'Distribution of interventions' Interpret as the relative cohort sizes in an experimental study Natural causation coef�cient: ^ P ( x ) = P ( x ) file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 9/67

  10. 5/2/2019 talk slides Causation coe�cient Causation coe�cient x 2 P ) 2 ^ ^ ^ V ar [ X ] = ∑ ( x ) − ( ∑ x P ( x ) x x y 2 ) 2 ^ ^ V a r X [ Y ] = ∑ ∑ P ( y | ) x ^ P ( x ) − ( ∑ ∑ yP ( y | ) x ^ P ( x ) ^ x y x y ^ ^ ^ ∑ x ∑ y xyP ( y | ) x ^ P ( x ) − ∑ x P x ( x ) ∑ x ∑ y yP ( y | ) x ^ P ( x ) γ X → Y = − − − − − − − − − − − − − ^ √ V ar [ X ] V a r X [ Y ] ^ file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 10/67

  11. 5/2/2019 talk slides Interpretation of Interpretation of γ - perfect positive/negative linear correlation ρ = ±1 - perfect positive/negative linear causation γ = ±1 - "linearly uncorrelated" ρ = 0 - "linearly acausal" γ = 0 file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 11/67

  12. 5/2/2019 talk slides No-confounding No-confounding implies P ( y ∣ x ) = P ( y ∣ x ^ ) γ X → Y = ρ Converse holds for Bernoulli (binary) random variables x y file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 12/67

  13. 5/2/2019 talk slides Independence and Invariance Independence and Invariance De�nitions: and are independent iff X Y P ( y ∣ x ) = P ( y ), ∀ x , y is invariant to iff Y X P ( y ∣ x ^ ) = P ( y ), ∀ x , y Lemmas: For Bernoulli , , iff and are independent X Y ρ = 0 X Y For Bernoulli , , iff is invariant to X Y γ X → Y = 0 Y X file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 13/67

  14. 5/2/2019 talk slides Average treatment e�ect Average treatment e�ect For Bernoulli random variables: ATE ( X → Y ) ≡ P ( Y = 1 ∣ do ( X = 1)) − P ( Y = 1 ∣ do ( X = 0)) − − − − − − − − ^ V ar [ X ] √ γ X → Y = ATE( X → Y ) V a r X [ Y ] ^ has the same sign as γ ATE( X → Y ) > 0 - treatment is more effective ATE( X → Y ) < 0 - treatment is less effective ATE( X → Y ) file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 14/67

  15. 5/2/2019 talk slides Plot causation vs correlation Plot causation vs correlation Every point on a plot is a structural causal model γρ file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 15/67

  16. 5/2/2019 talk slides file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 16/67

  17. 5/2/2019 talk slides Invariant and independent Invariant and independent Neither manipulation nor observation of changes/provides information about X Y e.g. Two events outside each other's past and future light cone file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 17/67

  18. 5/2/2019 talk slides file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 18/67

  19. 5/2/2019 talk slides Causation vs. correlation: common causation Causation vs. correlation: common causation "If an improbable coincidence has occurred, there must exist a common cause" (Reichenbach 1956) e.g. Myopia and ambient lighting at night (Quinn et al. 1999) file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 19/67

  20. 5/2/2019 talk slides Inverse causation Inverse causation and have the opposite sign ρ γ e.g. Tuberculosis in Arizona (Gardner 1982) file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 20/67

  21. 5/2/2019 talk slides Example model: inverse causation Example model: inverse causation Let and . The following model exhibits ϵ Z ∼ Bernoulli(1/2) ϵ Y ∼ Bernoulli(3/4) inverse causation: Z = ϵ Z X = Z Y = { ¬ Z if ϵ Y = 1 X if ϵ Y = 0 file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 21/67

  22. 5/2/2019 talk slides Inverse causation probability distributions Inverse causation probability distributions file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 22/67

  23. 5/2/2019 talk slides Causation vs. correlation: unfaithfulness Causation vs. correlation: unfaithfulness and are unfaithful if they are independent but not invariant X Y I de�ne this as a 'local' version of unfaithful distribution (Spirtes et al. 1993) file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 23/67

  24. 5/2/2019 talk slides "Friedman's thermostat" "Friedman's thermostat" Observe correlation between furnace and outside temperature Observe no correlation between furnace and inside temperature Observe no correlation between inside and outside temperature file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 24/67

  25. 5/2/2019 talk slides "Traitorous lieutenant" "Traitorous lieutenant" General wishes to send one bit, recipient XORs bits For 1 , send (0, 1) or (1, 0) with equal probability For 0 , send (1, 1) or (0, 0) with equal probability Lieutenant (loyal) General Recipient Lieutenant (traitor) file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 25/67

  26. 5/2/2019 talk slides Genuine causation and confounding bias Genuine causation and confounding bias and have the same sign ρ γ May be biased by confounders file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 26/67

  27. 5/2/2019 talk slides Recovering intuition: Why do we think correlation Recovering intuition: Why do we think correlation ≈ causation? causation? Need a way to analyze behavior of 'typical' models Don't draw samples from a model, draw models from a space of models How to parameterize that space? file:///home/josh/causal-programming-guest-lecture/slides.html?print-pdf#/ 27/67

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