Normalized Probabilistic Semantics is Not Associative

Abstract

Normalization, $D(X+1)\to D(X)+1$, fails to form a distributive law, forcing the composition of normalized stochastic kernels to be non-associative. We introduce a first normalized-by-construction probabilistic semantics: the non-associative monad - the magmad - of normalized distributions. Non-associativity allows this semantics to express both observations and interventions, both Pearl's and Jeffrey's updates, and both evidential and causal interpretations. Within this framework, we derive causality results including Pearl's front-door and back-door criteria.

40 pages, highlights do-notation, see v2 for string diagrammatic notation