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The Observational Partial Order of Causal Structures with Latent Variables

Marina Maciel Ansanelli, Elie Wolfe, Robert W. Spekkens

TL;DR

This work introduces the observational partial order on causal structures with latent variables, focusing on how observational distributions over visible variables distinguish competing structures. It provides a complete characterization for the partial order among 3-node mDAGs and a partial solution for 4-node cases, using a toolkit of dominance/nondominance rules (including HLP Edge-Adding and progressive Facet-Merging) and the analysis of inequality constraints. A key finding is that nonalgebraic (inequality-constraint) causal structures are increasingly ubiquitous as the number of visible variables grows, implying widespread potential quantum-classical gaps and signaling the inadequacy of CI-only methods for causal discovery in larger graphs. The results underscore the importance of inequality and nested Markov constraints for effective causal discovery and model selection, and they provide a foundation for identifying which observational equivalence classes are uniquely identifiable from data. The work also connects to quantum causality by highlighting when classical models can be decisively ruled out without fine-tuning, based on inequality constraints.

Abstract

For two causal structures with the same set of visible variables, one is said to observationally dominate the other if the set of distributions over the visible variables realizable by the first contains the set of distributions over the visible variables realizable by the second. Knowing such dominance relations is useful for adjudicating between these structures given observational data. We here consider the problem of determining the partial order of equivalence classes of causal structures with latent variables relative to observational dominance. We provide a complete characterization of the dominance order in the case of three visible variables, and a partial characterization in the case of four visible variables. Our techniques also help to identify which observational equivalence classes have a set of realizable distributions that is characterized by nontrivial inequality constraints, analogous to Bell inequalities and instrumental inequalities. We find evidence that as one increases the number of visible variables, the equivalence classes satisfying nontrivial inequality constraints become ubiquitous. (Because such classes are the ones for which there can be a difference in the distributions that are quantumly and classically realizable, this implies that the potential for quantum-classical gaps is also ubiquitous.) Furthermore, we find evidence that constraint-based causal discovery algorithms that rely solely on conditional independence constraints have a significantly weaker distinguishing power among observational equivalence classes than algorithms that go beyond these (i.e., algorithms that also leverage nested Markov constraints and inequality constraints).

The Observational Partial Order of Causal Structures with Latent Variables

TL;DR

This work introduces the observational partial order on causal structures with latent variables, focusing on how observational distributions over visible variables distinguish competing structures. It provides a complete characterization for the partial order among 3-node mDAGs and a partial solution for 4-node cases, using a toolkit of dominance/nondominance rules (including HLP Edge-Adding and progressive Facet-Merging) and the analysis of inequality constraints. A key finding is that nonalgebraic (inequality-constraint) causal structures are increasingly ubiquitous as the number of visible variables grows, implying widespread potential quantum-classical gaps and signaling the inadequacy of CI-only methods for causal discovery in larger graphs. The results underscore the importance of inequality and nested Markov constraints for effective causal discovery and model selection, and they provide a foundation for identifying which observational equivalence classes are uniquely identifiable from data. The work also connects to quantum causality by highlighting when classical models can be decisively ruled out without fine-tuning, based on inequality constraints.

Abstract

For two causal structures with the same set of visible variables, one is said to observationally dominate the other if the set of distributions over the visible variables realizable by the first contains the set of distributions over the visible variables realizable by the second. Knowing such dominance relations is useful for adjudicating between these structures given observational data. We here consider the problem of determining the partial order of equivalence classes of causal structures with latent variables relative to observational dominance. We provide a complete characterization of the dominance order in the case of three visible variables, and a partial characterization in the case of four visible variables. Our techniques also help to identify which observational equivalence classes have a set of realizable distributions that is characterized by nontrivial inequality constraints, analogous to Bell inequalities and instrumental inequalities. We find evidence that as one increases the number of visible variables, the equivalence classes satisfying nontrivial inequality constraints become ubiquitous. (Because such classes are the ones for which there can be a difference in the distributions that are quantumly and classically realizable, this implies that the potential for quantum-classical gaps is also ubiquitous.) Furthermore, we find evidence that constraint-based causal discovery algorithms that rely solely on conditional independence constraints have a significantly weaker distinguishing power among observational equivalence classes than algorithms that go beyond these (i.e., algorithms that also leverage nested Markov constraints and inequality constraints).

Paper Structure

This paper contains 42 sections, 31 theorems, 34 equations, 32 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Explaining Observed Data with a Causal Structure
  4. Observational Equivalence and Dominance
  5. mDAGs
  6. Objectives and Methods
  7. Restriction of scope to sets of mDAGs consistent with a fixed nodal ordering
  8. Special Cases: Confounder-free and Directed-edge-free mDAGs
  9. Full Solution of Observational Partial Order for 3-node mDAGs
  10. Significance of observational partial order for causal discovery
  11. Model selection for probability distributions
  12. Model selection for finite-run samples of probability distributions
  13. Attesting to the presence of inequality constraints
  14. How to Express Partial Information about the Observational Equivalence Partition
  15. Rules for Establishing Observational Dominance
  16. ...and 27 more sections

Key Result

Theorem 1

Let $\mathcal{G}$ be a pDAG, and associate it to a classical unrestricted semantics under arbitrary cardinalities of the latent variables. Let $A$, $B$ and $C$ be three disjoint subsets of the set of visible nodes, $\mathtt{Vnodes}(\cal G)$. Then:

Figures (32)

  • Figure 1.1: Two observationally equivalent causal structures.
  • Figure 2.1: An example of nonalgebraic pDAGs that are observationally equivalent. The first of this triple is generally known as the instrumental scenario.
  • Figure 2.2: (a) A pDAG. (b) The pDAG obtained from (a) by exogenizing the latent node $\alpha$. (c) The pDAG obtained from that of (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. The pDAG in (c) is RE-reduced.
  • Figure 2.3: (a) A RE-reduced pDAG. (b) The mDAG associated with (a). This mDAG has simplicial complex $\mathcal{B}=\{\{a\},\{b\},\{d\},\{a,b\},\{a,d\},\{b,d\},\{a,b,d\}\}$. This is the mDAG associated with all of the pDAGs of Fig. \ref{['fig_example_lemmas']}. The inclusion-maximal element of $\mathcal{B}$ are indicated by the red loop on the mDAG.
  • Figure 2.4: (a) mDAG with trivial directed structure and with simplicial complex $\mathcal{B}=\{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\}$. (b) mDAG with trivial directed structure and with simplicial complex $\mathcal{B'}=\{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$.The facets (inclusion-maximal elements of $\mathcal{B}$ and $\mathcal{B'}$) are indicated by red loops.
  • ...and 27 more figures

Theorems & Definitions (66)

  • Definition 1: Children, Parents, Descendants, Ancestors
  • Definition 2: Subgraph
  • Definition 3: Partitioned DAG
  • Definition 4: Set of pDAGs consistent with a fixed nodal Ordering
  • Definition 5: Set of probability distributions with cardinality vector $\vec{c}_\text{vis}$ realizable by a pDAG by a causal model with classical unrestricted semantics under arbitrary cardinality of the latent variables
  • Definition 6: Observational profile
  • Definition 7: Algebraicness
  • Theorem 1: d-separation and conditional independence
  • Definition 8: Observational dominance and equivalence of pDAGs
  • Lemma 1: Exogenize Latent Nodes
  • ...and 56 more