Benefit Probability Bounds

Abstract

The two Boolean potential-outcome marginals give algebraic bounds on the benefit mass of every normalized nonnegative joint law.

Theorem 1.1 (Potential-outcome marginals bound the benefit probability).

$$\forall mass\in \mathrm{Prod}\left(Bool,Bool\right)\to Real,\left(\forall pair\in \mathrm{Prod}\left(Bool,Bool\right),0\le mass\left(pair\right)\wedge mass\left(\mathrm{pair}\left(false,false\right)\right)+mass\left(\mathrm{pair}\left(false,true\right)\right)+mass\left(\mathrm{pair}\left(true,false\right)\right)+mass\left(\mathrm{pair}\left(true,true\right)\right)=1\right)\Rightarrow \left(\max \left(0,mass\left(\mathrm{pair}\left(false,true\right)\right)+mass\left(\mathrm{pair}\left(true,true\right)\right)-\left(mass\left(\mathrm{pair}\left(true,false\right)\right)+mass\left(\mathrm{pair}\left(true,true\right)\right)\right)\right)\le mass\left(\mathrm{pair}\left(false,true\right)\right)\wedge mass\left(\mathrm{pair}\left(false,true\right)\right)\le \min \left(mass\left(\mathrm{pair}\left(false,true\right)\right)+mass\left(\mathrm{pair}\left(true,true\right)\right),1-\left(mass\left(\mathrm{pair}\left(true,false\right)\right)+mass\left(\mathrm{pair}\left(true,true\right)\right)\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Causal/BenefitProbabilityBounds.benefit_probability_bounds (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let mass be a normalized nonnegative joint law of the Boolean pair of potential outcomes. The benefit probability is the mass of the false-true response type.

The treatment-one marginal is the sum of the false-true and true-true masses. The treatment-zero marginal is the sum of the true-false and true-true masses.

Nonnegativity of the true-false cell gives the lower marginal-difference bound. Nonnegativity of the true-true and false-false cells gives the two upper bounds.

References

• Truth anchor: D5/S3/ConceptDynamics/Causal/BenefitProbabilityBounds.benefit_probability_bounds
• Dependency: D5/S3/ConceptDynamics/Causal/PrincipalStrata