Probabilistic Reasoning Across the Causal Hierarchy

Abstract

We propose a formalization of the three-tier causal hierarchy of association, intervention, and counterfactuals as a series of probabilistic logical languages. Our languages are of strictly increasing expressivity, the first capable of expressing quantitative probabilistic reasoning—including conditional independence and Bayesian inference—the second encoding do-calculus reasoning for causal effects, and the third capturing a fully expressive do-calculus for arbitrary counterfactual queries. We give a corresponding series of finitary axiomatizations complete over both structural causal models and probabilistic programs, and show that satisfiability and validity for each language are decidable in polynomial space.

Thinking About Causation: A Causal Language with Epistemic Operators

“…Causal Team Semantics was proposed with the intention of supporting languages that discuss both accidental and causal dependencies. This is a topic that has gained quite some interest in recent years (see, e.g., [12,21]). Causal Team Semantics was developed along the lines of a non-modal tradition of logics of dependence and independence (e.g.…”

“…Then, while the fourth case in the second column takes K outside the scope of [ #-X= #x ], cases six through eight 'push' [γ ′ ] inside the formula until it has only an atom Z=z in front, at which moment [γ ′ ] is eliminated (fifth case). 21 The ninth case deals with nested announcements following an 'inside-first' strategy. 20 Recall that Z=z is the particular case of [ #-X= #x ]Z=z where #-X is empty.…”

“…20 Recall that Z=z is the particular case of [ #-X= #x ]Z=z where #-X is empty. 21 Proving that the translation ends and that announcement operators are indeed eventually eliminated requires some care. The crucial thing to notice is that, in cases sixth through eighth, the formula occurring under the scope of announcement operators on the right-hand side is less complex that the one occurring under the scope of the same announcement operator on the lefthand side.…”

“…Even though the standard logical approach to causal reasoning ( [23,17,18]) can model epistemic uncertainty 1 , it does not permit reasoning about the interaction between causal and epistemic reasoning in the object language. Although recently there have been proposals adding probabilistic expressions to the object language (e.g., [21]), very little has been done on combining causal and qualitative epistemic reasoning. 2 However, this kind of reasoning occurs frequently in our daily life, especially in connection with counterfactual thinking.…”

Observing Interventions: A logic for thinking about experiments

“…Even though the standard logical approach to causal reasoning (Pearl 2009;Halpern 2000Halpern , 2016 can model epistemic uncertainty 1 , it does not come with an object language that can make statements about the epistemic state of some agent. There are recent proposals adding probabilistic expressions to the object language (e.g., Ibeling and Icard 2020), but very little has been done on combining causal and qualitative epistemic reasoning. 2 However, this kind of reasoning occurs frequently in our daily life.…”