Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interpretable Causal Graphical Models for Equilibrium Systems with Confounding

Kai Z. Teh, Kayvan Sadeghi, Terry Soo

Abstract

In applications, quantities of interest are often modelled in equilibrium or an equilibrium solution is sought. The presence of confounding makes causal inference in this setting challenging. We provide interpretable graphical models for equilibrium systems with confounding using anterial graphs (Lauritzen and Sadeghi, 2018), a class of graphs containing directed acyclic graphs, ancestral graphs, and chain graphs. In this setting, we provide valid graphical representations of both counterfactual variables and observational variables, which we relate to counterfactual graphs (Shpitser and Pearl, 2007) and single-world intervention graphs (Richardson and Robins,2013). As an application of this graphical representation, we provide an element-wise procedure of selecting adjustment sets that flexibly include and exclude given covariates.

Interpretable Causal Graphical Models for Equilibrium Systems with Confounding

Abstract

In applications, quantities of interest are often modelled in equilibrium or an equilibrium solution is sought. The presence of confounding makes causal inference in this setting challenging. We provide interpretable graphical models for equilibrium systems with confounding using anterial graphs (Lauritzen and Sadeghi, 2018), a class of graphs containing directed acyclic graphs, ancestral graphs, and chain graphs. In this setting, we provide valid graphical representations of both counterfactual variables and observational variables, which we relate to counterfactual graphs (Shpitser and Pearl, 2007) and single-world intervention graphs (Richardson and Robins,2013). As an application of this graphical representation, we provide an element-wise procedure of selecting adjustment sets that flexibly include and exclude given covariates.

Paper Structure

This paper contains 24 sections, 17 theorems, 24 equations, 18 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Probabilistic Graphs
  4. Interventions and Confounder Selection
  5. Causal Interpretation of Chain Graphs
  6. Structured Equilibrium Models with Confounding
  7. Interpreting the constructed graphical model $\mathcal{G}$
  8. Examples and simulations
  9. Graphical Interventions of Structural Equilibrium Models
  10. Representing interventions graphically
  11. Counterfactual Interpretations
  12. Representing the coupling graphically
  13. Single-world interpretation
  14. Confounder Selection with Constraints
  15. Conclusion and Future Work
  16. ...and 9 more sections

Key Result

Theorem 1

Let $P$ be the joint distribution over the variables $X_V$ induced from a structural equilibrium model and let the corresponding graph $\mathcal{G}$ over the nodes $V$, as constructed in Algorithm obsalg, be anterial. If $P$ is a compositional graphoid, then $P$ is Markovian to $\mathcal{G}$.

Figures (18)

  • Figure 1: Chain component $\tau = \{3,4,5\}$ and parents $\textnormal{pa}(\tau) = \{1,2\}$. For every node $i\in \tau$, Algorithm \ref{['obsalg']} constructs $\textnormal{ne}(i)$ and $\textnormal{pa}(i)$ such that the conditional marginal $J^{\tau}_i(\cdot \,|\, x_{\tau\backslash i}; x_{\textnormal{pa}(\tau)})$ does not depend on nodes outside of $\textnormal{pa}(i)\cup \textnormal{ne}(i)$ (formalised as Lemma \ref{['localind']} in the Proofs Section).
  • Figure 2: Left: Chain-connected anterial graph $\mathcal{G}$. Middle: Interpretation of bidirected edges $3\leftrightarrow 4$ and $2\leftrightarrow 4$ in $\mathcal{G}$ as unobserved latent variables $U_1$ and $U_2$ respectively, from the chain graph $\mathcal{G}'$. Right: Graph $\alpha_{\textnormal{m}}(\mathcal{G}';\{U_1,U_2\})$ obtained by marginalising $\mathcal{G}'$ over the latent variables $U_1$ and $U_2$.
  • Figure 3: Left: Chain-connected anterial graph $\mathcal{G}$. Middle: Interpretation of the undirected edge $2-3$ as the selection bias variable $L_1$ from the ancestral graph $\mathcal{G}'$. Right: Graph $\alpha_{\textnormal{c}}(\mathcal{G}';L_1)$ obtained by conditioning $\mathcal{G}'$ over the selection bias variable $L_1$.
  • Figure 4: Left: Corresponding chain-connected anterial graph $\mathcal{G}_1$. Middle: Structural equilibrium model. Conditioning on $5$, the distribution $J^{\{5,6\}}_6(\cdot \,|\, x_{5};x_{\{2,4\}})$ does not depend on $x_{\{2,4\}}$, thus $\textnormal{pa}(6)=\emptyset$. Right: Distribution and coupling of the error variables.
  • Figure 5: Left: Corresponding chain-connected anterial graph $\mathcal{G}_2$. Middle: Structural equilibrium model. Conditioning on $3$, the distribution $J^{\{3,4\}}_4(\cdot \,|\, x_{3};x_{\{1,2\}})$ depends on $x_{2}$ only, thus $\textnormal{pa}(4)=2$, similarly we have $\textnormal{pa}(3)=1$. Right: Distribution and coupling of the error variables.
  • ...and 13 more figures

Theorems & Definitions (58)

  • Definition 1: Anterial graphs
  • Remark 1
  • Definition 2: Maximal graphs
  • Definition 3: Markov property
  • Definition 4: Faithfulness
  • Definition 5: Compositional graphoid
  • Remark 2
  • Definition 6: Structural causal model (SCM)
  • Definition 7: Structural equilibrium model
  • Remark 3
  • ...and 48 more