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Everything that can be learned about a causal structure with latent variables by observational and interventional probing schemes

Marina Maciel Ansanelli, Elie Wolfe, Robert W. Spekkens

TL;DR

This paper addresses identifiability limits for causal structures with latent variables under various probing schemes. It introduces mDAGs as the fundamental objects capturing indistinguishability across causal explanations and proves that, under informationally complete probing (Observe&Do) and even under all-patterns or one-value do-interventions, equivalence and dominance of pDAGs are completely characterized by structural dominance in the corresponding mDAGs. The results unify prior observations, showing that even weakened schemes retain the same discriminative power with respect to mDAGs, and provide a concrete partial order for causal structures with three or four visible nodes. The work also connects full-SWIGs to 3-mDAGs and discusses open questions, including extensions to edge interventions, weaker probing schemes, and quantum latent variables.

Abstract

What types of differences among causal structures with latent variables are impossible to distinguish by statistical data obtained by probing each visible variable? If the probing scheme is simply passive observation, then it is well-known that many different causal structures can realize the same joint probability distributions. Even for the simplest case of two visible variables, for instance, one cannot distinguish between one variable being a causal parent of the other and the two variables sharing a latent common cause. However, it is possible to distinguish between these two causal structures if we have recourse to more powerful probing schemes, such as the possibility of intervening on one of the variables and observing the other. Herein, we address the question of which causal structures remain indistinguishable even given the most informative types of probing schemes on the visible variables. We find that two causal structures remain indistinguishable if and only if they are both associated with the same mDAG structure (as defined by Evans (2016)). We also consider the question of when one causal structure dominates another in the sense that it can realize all of the joint probability distributions that can be realized by the other using a given probing scheme. (Equivalence of causal structures is the special case of mutual dominance.) Finally, we investigate to what extent one can weaken the probing schemes implemented on the visible variables and still have the same discrimination power as a maximally informative probing scheme.

Everything that can be learned about a causal structure with latent variables by observational and interventional probing schemes

TL;DR

This paper addresses identifiability limits for causal structures with latent variables under various probing schemes. It introduces mDAGs as the fundamental objects capturing indistinguishability across causal explanations and proves that, under informationally complete probing (Observe&Do) and even under all-patterns or one-value do-interventions, equivalence and dominance of pDAGs are completely characterized by structural dominance in the corresponding mDAGs. The results unify prior observations, showing that even weakened schemes retain the same discriminative power with respect to mDAGs, and provide a concrete partial order for causal structures with three or four visible nodes. The work also connects full-SWIGs to 3-mDAGs and discusses open questions, including extensions to edge interventions, weaker probing schemes, and quantum latent variables.

Abstract

What types of differences among causal structures with latent variables are impossible to distinguish by statistical data obtained by probing each visible variable? If the probing scheme is simply passive observation, then it is well-known that many different causal structures can realize the same joint probability distributions. Even for the simplest case of two visible variables, for instance, one cannot distinguish between one variable being a causal parent of the other and the two variables sharing a latent common cause. However, it is possible to distinguish between these two causal structures if we have recourse to more powerful probing schemes, such as the possibility of intervening on one of the variables and observing the other. Herein, we address the question of which causal structures remain indistinguishable even given the most informative types of probing schemes on the visible variables. We find that two causal structures remain indistinguishable if and only if they are both associated with the same mDAG structure (as defined by Evans (2016)). We also consider the question of when one causal structure dominates another in the sense that it can realize all of the joint probability distributions that can be realized by the other using a given probing scheme. (Equivalence of causal structures is the special case of mutual dominance.) Finally, we investigate to what extent one can weaken the probing schemes implemented on the visible variables and still have the same discrimination power as a maximally informative probing scheme.
Paper Structure (13 sections, 14 theorems, 50 equations, 20 figures)

This paper contains 13 sections, 14 theorems, 50 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Full-SWIGs
  4. Probing Schemes and Shadows
  5. Equivalence and Dominance of causal structures under the Observe&Do Probing scheme
  6. Equivalence and Dominance under all patterns of do-interventions
  7. One-Value do-interventions
  8. Structural Partial Order for mDAGs with three or four visible nodes
  9. Confounder-Free and Directed-Edge-Free mDAGs
  10. Conclusion
  11. Proof of Lemma \ref{['lemma_commutation_maps']}
  12. Proof of the Case 2 of Theorem \ref{['th_do_and_ob']}
  13. Proof of Proposition \ref{['theorem_confounder_free']}

Key Result

Lemma 1

Let $\mathcal{G}$ be a pDAG, and let ${\tt EndoLNodes}(\mathcal{G})\subseteq\mathtt{Lnodes}(\mathcal{G})$ be the set of latent nodes of $\mathcal{G}$ that are endogenous (i.e., they have one or more parents in $\mathcal{G}$). Construct the pDAG ${\tt Exog}(\mathcal{G})$ as follows. For every $u\in {

Figures (20)

  • Figure 2.1: (a) A pDAG, with visible nodes in white and latent nodes in gray. (b) The pDAG obtained from (a) by exogenizing the latent node $\alpha$. (c) The pDAG obtained from (b) by removing the latent node $\alpha$, which is redundant to $\beta$ since the children of $\alpha$ are a subset of the children of $\beta$. By Lemmas \ref{['lemma_exogenize_latents']} and \ref{['lemma_remove_redundant_latents']}, these three pDAGs are observationally equivalent. Because no further exogenization or removal of redundant latents is possible, the pDAG in (c) is the RE-reduced pDAG for the pDAG in (a).
  • Figure 2.2: (a) The mDAG $(\mathcal{D},\mathcal{B})$ where the directed structure $\mathcal{D}$ is trivial and the simplicial complex is $\mathcal{B}=\{\{0\},\{1\},\{2\},\{0,1\},\{0,2\},\{1,2\},\{0,1,2\}\}$. (b) The mDAG $(\mathcal{D}',\mathcal{B}')$ where the directed structure $\mathcal{D}'$ is again trivial and the simplicial complex is $\mathcal{B'}=\{\{0\},\{1\},\{2\},\{0,1\},\{0,2\},\{1,2\}\}$. Because $\mathcal{D}'\subset \mathcal{D}$ and $\mathcal{B}'\subset \mathcal{B}$, it follows from Definition \ref{['def_structural_dominance']} that (a) structurally dominates (b). The facets (inclusion-maximal elements of $\mathcal{B}$ and $\mathcal{B'}$) are indicated by the red loops.
  • Figure 3.1: (a) The pDAG Cause-Effect&Confounder. (b) The SWIG that represents the possibility of intervening on the node $a$ of Cause-Effect&Confounder.
  • Figure 3.2: Example of the operation of splitting all visible nodes of a pDAG $\mathcal{G}$ to obtain the corresponding full-SWIG $\mathscr{G}=\mathtt{split}(\mathcal{G})$.
  • Figure 3.3: Example of the operation of splitting all nodes of an mDAG $\mathfrak{G}$ to obtain the corresponding 3-mDAG $\mathbb{G}=\mathtt{split}(\mathfrak{G})$.
  • ...and 15 more figures

Theorems & Definitions (33)

  • Definition 1: Partitioned DAG
  • Definition 2: Observational dominance and equivalence of pDAGs
  • Lemma 1: Exogenize Latent Nodes
  • Lemma 2: Eliminate Redundant Latent Nodes
  • Lemma 3
  • Definition 3: Simplicial complex
  • Definition 4: The map $\mathtt{LnodesToFaces}$ taking pDAGs to mDAGs
  • Definition 5: Canonical pDAG associated with an mDAG
  • Definition 6: Structural Dominance relation between mDAGs
  • Lemma 4
  • ...and 23 more