Multivariate Counterfactual Systems And Causal Graphical Models

Abstract

Among Judea Pearl's many contributions to Causality and Statistics, the graphical d-separation criterion, the do-calculus and the mediation formula stand out. In this chapter we show that d-separation provides direct insight into an earlier causal model originally described in terms of potential outcomes and event trees. In turn, the resulting synthesis leads to a simplification of the do-calculus that clarifies and separates the underlying concepts, and a simple counterfactual formulation of a complete identification algorithm in causal models with hidden variables.

Potential Outcome and Decision Theoretic Foundations for Statistical Causality

“…Consequently Propositions 45 and 46 and Theorem 65(c) inRichardson and Robins (2013) are incorrect. Correct reformulations are given below in Theorems 10, 11 and 12.10 This is as to be expected since d-separation encodes the global property that is implied by the local property.11 Note that in other papers(Shpitser et al, 2021;Malinsky et al, 2019;Richardson and Robins, 2013) d-connection for SWIGs is defined such that fixed nodes may never occur as non-endpoint vertices on d-connecting paths. In those papers we never formally condition on fixed nodes.…”

“…from fixed nodes a i that correspond to vertices A i that are not parents of V i in G given the parents of X i (a) in G(a), both random and fixed; see Shpitser et al (2021), Malinsky et al (2019), Robins (2018). Specifically we have:…”

A Formal Causal Interpretation of the Case-Crossover Design

“…In the absence of rare disease and under the stronger constant effects assumption that 𝛽 = 𝜆 1 𝑣𝑘 ( ū𝑘 )∕𝜆 0 𝑣𝑘 ( ū𝑘 ) does not depend on 𝑣, 𝑘, or ū𝑘 , it follows from (A.4) that we can expand the multiplicative bias as 4) holds under the assumption (which we assume is true) that the causal DAG in Figure 1 represents an underlying FFRCISTG counterfactual causal model (Robins, 1986) and thus also under Pearl's NPSEM with independent errors. See Richardson and Robins (2013) and Shpitser, Richardson, and Robins (2020).…”