Causality Actions, Confounders and Interventions Christos Dimitrakakis October 30, 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 1 / 22
Introduction Introduction Decision diagrams Common structural assumptions Interventions Policy evaluation and optimisation Individual effects and counterfactuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 2 / 22
Introduction Headaches and aspirins Example 1 (Population effects) 1 High dose Cured 1 Low dose Side-effects 0 . 8 0 . 8 Response Response 0 . 6 0 . 6 0 . 4 0 . 4 0 . 2 0 . 2 0 0 − 2 − 1 0 1 2 0 1 2 3 4 5 Dose Sensitivity (a) Dose-response curve. (b) Response distribution Figure: Investigation the response of the population to various doses of the drug. ▶ Is aspirin an effective cure for headaches? ▶ Does having a headache lead to aspirin-taking? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 3 / 22
Introduction Example 2 (Individual effects) ▶ Effects of Causes: Will my headache pass if I take an aspirin? ▶ Causes of Effects: Would my headache have passed if I had not taken an aspirin? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 4 / 22
Introduction Overview Inferring causal models We can distinguish different models from observational or experimental data. Inferring individual effects The effect of possible intervention on an individual is not generally determinable. We usually require strong assumptions. Decision-theoretic view There are many competing approaches to causality. We will remain within the decision-theoretic framework, which allows us to crisply define both our knowledge and assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 5 / 22
Introduction What causes what? Example 3 θ θ a t x t a t x t (a) Independence of a t . (b) Independence of x t . Suppose we have data x t , a t where ▶ x t : lung cancer ▶ a t : smoking Does smoking cause lung cancer or does lung cancer make people smoke? Can we compare the two models above to determine it? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 6 / 22
Introduction What causes what? Example 3 θ θ a t x t a t x t (a) Independence of a t . (b) Independence of x t . Suppose we have data x t , a t where ▶ x t : lung cancer ▶ a t : smoking Does smoking cause lung cancer or does lung cancer make people smoke? Can we compare the two models above to determine it? ∏ ∏ ∏ P θ ( D ) = P θ ( x t , a t ) = P θ ′ ( x t | a t ) P θ ′ ( a t ) = P θ ′′ ( a t | x t ) P θ ′′ ( x t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t t t C. Dimitrakakis Causality October 30, 2019 6 / 22
Introduction Decision diagrams y t θ x t a t Figure: A typical decision diagram where x t : individual information, y t : individual result, a t : action, π : policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 7 / 22
Introduction Decision diagrams y t θ U x t a t Figure: A typical decision diagram where x t : individual information, y t : individual result, a t : action, π : policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 7 / 22
Introduction Decision diagrams y t θ U x t a t π Figure: A typical decision diagram where x t : individual information, y t : individual result, a t : action, π : policy Example 4 (Taking an aspirin) ▶ Individual t ▶ Individual information x t ▶ a t = 1 if t takes an aspirin, and 0 otherwise. ▶ y t = 1 if the headache is cured in 30 minutes, 0 otherwise. ▶ π : intervention policy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 7 / 22
Introduction Decision diagrams y t θ U x t a t π Figure: A typical decision diagram where x t : individual information, y t : individual result, a t : action, π : policy Example 4 (A recommendation system) ▶ x t : User information (random variable) ▶ a t : System action (random variable) ▶ y t : Click (random varaible) ▶ π : recommendation policy (decision variable). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 7 / 22
Introduction Decision diagrams Conditional distributions and decision variables. P ( A | B ) ≜ P ( A ∩ B ) . P ( B ) The conditional distribution of decisions π ( a ) ≡ P π ( a ) ≡ P ( a | π ) . P π θ ( a ) ≡ P ( a | θ, π ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 8 / 22
Introduction Common structural assumptions Basic causal structures Non-cause a t y t π Figure: π does not cause y No confounding y t π a t Figure: No confounding: π causes y t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 9 / 22
Introduction Common structural assumptions Basic causal structures Non-cause θ a t y t π Figure: π does not cause y No confounding θ y t π a t Figure: No confounding: π causes y t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 9 / 22
Introduction Common structural assumptions Covariates Sufficient covariate x t a t y t π Figure: Sufficient covariate x t Instrumental variables and confounders z t x t a t y t π Figure: Instrumental variable z t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 10 / 22
Introduction Common structural assumptions Covariates Sufficient covariate x t θ a t y t π Figure: Sufficient covariate x t Instrumental variables and confounders z t x t θ a t y t π Figure: Instrumental variable z t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 10 / 22
Interventions Modelling interventions ▶ Observational data D . ▶ Policy space Π . Default policy The space of policies Π includes a default policy π 0 , under which the data was collected. Intervention policies Except π 0 , policies π ∈ Π represent different interventions specifying a distribution π ( a t | x t ). ▶ Direct interventions. ▶ Indirect interventions and non-compliance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 11 / 22
Interventions Example 5 (Weight loss) y t θ U x t a t π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 12 / 22
Interventions Example 5 (Weight loss) x t y t θ U z t a t π Figure: Model of non-compliance as a confounder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 12 / 22
Policy evaluation and optimisation The value of an observed policy y t θ U x t a t π Figure: Basic decision diagram ˆ a ∗ ˆ D ∈ arg max E D ( U | a ) , a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 13 / 22
Policy evaluation and optimisation The value of an observed policy y t θ U x t a t π Figure: Basic decision diagram 1 ∑ ˆ E D ( U | a ) ≜ U ( a t , y t ) (3.1) | { t | a t = a } | t : a t = a ≈ E π 0 ( a t , y t ) ∼ P π 0 θ ( U | a ) θ . (3.2) ˆ a ∗ ˆ D ∈ arg max E D ( U | a ) , a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dimitrakakis Causality October 30, 2019 13 / 22
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