Is Causal Reasoning Harder Than Probabilistic Reasoning?

Abstract

Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. Along the way we answer several open problems concerning the complexity of well-known probability logics, in particular demonstrating the ${\exists \mathbb {R}}$ -completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability.

Probabilistic Reasoning Across the Causal Hierarchy

“…it suffices to show that SAT ind is ∃R-hard and that SAT poly is in ∃R. To show the former, we extend an argument given by Mossé et al (2022), and to show the latter, we repeat the proof given by Mossé et al (2022) (see also Ibeling and Icard (2020)).…”

“…], recursing to define a semantics for all of L. Over the class of all (recursive, possibly infinite) SCMs, L has been axiomatized [13] by a set of principles called AX 3 , and the complexity of its satisfiability problem has been shown complete for the class ∃R [25]. The class of simple probability distributions over the atoms of L base is axiomatized by principles known as AX 1 [13], which we will abbreviate AX.…”

Interleaving Logic and Counting

“…This echoes a broader theme that reasoning about conditional probabilities already amounts to general reasoning about real fields [63, 93].…”