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Representation of Context-Specific Causal Models with Observational and Interventional Data

Eliana Duarte, Liam Solus

Abstract

We address the problem of representing context-specific causal models based on both observational and experimental data collected under general (e.g. hard or soft) interventions by introducing a new family of context-specific conditional independence models called CStrees. This family is defined via a novel factorization criterion that allows for a generalization of the factorization property defining general interventional DAG models. We derive a graphical characterization of model equivalence for observational CStrees that extends the Verma and Pearl criterion for DAGs. This characterization is then extended to CStree models under general, context-specific interventions. To obtain these results, we formalize a notion of context-specific intervention that can be incorporated into concise graphical representations of CStree models. We relate CStrees to other context-specific models, showing that the families of DAGs, CStrees, labeled DAGs and staged trees form a strict chain of inclusions. We end with an application of interventional CStree models to a real data set, revealing the context-specific nature of the data dependence structure and the soft, interventional perturbations.

Representation of Context-Specific Causal Models with Observational and Interventional Data

Abstract

We address the problem of representing context-specific causal models based on both observational and experimental data collected under general (e.g. hard or soft) interventions by introducing a new family of context-specific conditional independence models called CStrees. This family is defined via a novel factorization criterion that allows for a generalization of the factorization property defining general interventional DAG models. We derive a graphical characterization of model equivalence for observational CStrees that extends the Verma and Pearl criterion for DAGs. This characterization is then extended to CStree models under general, context-specific interventions. To obtain these results, we formalize a notion of context-specific intervention that can be incorporated into concise graphical representations of CStree models. We relate CStrees to other context-specific models, showing that the families of DAGs, CStrees, labeled DAGs and staged trees form a strict chain of inclusions. We end with an application of interventional CStree models to a real data set, revealing the context-specific nature of the data dependence structure and the soft, interventional perturbations.

Paper Structure

This paper contains 24 sections, 12 theorems, 76 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. CStrees
  4. CStree models
  5. Markov properties of CStrees
  6. Conditional independence models
  7. Context-specific conditional independence models
  8. Model equivalence
  9. Interventional CStrees
  10. Interventional CStrees
  11. $\mathcal{I}$-Markov properties of CStrees
  12. Markov equivalence of interventional CStrees
  13. Real Data Example
  14. Context fear conditioning data set description
  15. Model selection process
  16. ...and 9 more sections

Key Result

Theorem 3.3

Let $\mathbb{D}$, $\mathbb{C}$, $\mathbb{L}$ and $\mathbb{S}$ denote the collections of DAG models, CStree models, LDAG models and staged tree models, respectively. Then

Figures (15)

  • Figure 1: An LDAG that is not a CStree and a staged tree that is not an LDAG
  • Figure 2: A CStree representation of the context-specific conditional independence model $\mathcal{D}$ on four binary variables from Example \ref{['ex:CSI causal model']}.
  • Figure 3: The minimal context graphs of the CStree in Figure \ref{['fig:cstree']}.
  • Figure 4: The staged tree representation of two CStrees that are Markov equivalent to $\mathcal{T}$ from Figure \ref{['fig:cstree']}. The staged tree representation explicitly shows the models satisfy the CStree factorization \ref{['eqn:CStreefactorization']}. The minimal context graphs, described in Example \ref{['ex: equivalent CStrees']} show model equivalence much more easily via Corollary \ref{['cor: VP generalization']}.
  • Figure 5: The minimal context graphs of the CStree in Figure \ref{['fig:cstree']}.
  • ...and 10 more figures

Theorems & Definitions (46)

  • Example 1.1
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • Example 3.4
  • Remark 3.5
  • Lemma 3.6
  • Definition 3.7
  • Definition 3.8
  • Definition 3.9
  • ...and 36 more