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Abstracting Causal Models
Sander Beckers, Joseph Y. Halpern
TL;DR
The paper asks how to formalize when a high-level causal model faithfully abstracts a low-level one, particularly when micro-variables are combined into macro-variables. It develops increasingly restrictive transformation and abstraction notions, showing that such combinations fit constructive abstraction while identifying cases where stronger requirements are too demanding.
Problem
The paper examines whether high-level causal models faithfully abstract low-level models and argues that exact transformation can treat too many model differences as inessential.
Method
The paper develops a sequence from exact and uniform transformations to abstraction, strong abstraction, and constructive abstraction, with intervention mappings increasingly constrained by state mappings and potential interventions.
Results
Procedures that combine micro-variables into macro-variables are instances of constructive abstraction, and the examples considered by Rubenstein et al. are also covered.
Takeaways & Limitations
Constructive abstraction is identified as the notion likely to be most useful in practice, while weaker abstraction notions remain of interest.
Takeaways & Limitations
Strong abstraction can be too demanding when meaningful models fail its additional intervention requirements, including cases with logically dependent high-level variables.
Abstract
from arXiv · showhide
We consider a sequence of successively more restrictive definitions of abstraction for causal models, starting with a notion introduced by Rubenstein et al. (2017) called exact transformation that applies to probabilistic causal models, moving to a notion of uniform transformation that applies to deterministic causal models and does not allow differences to be hidden by the "right" choice of distribution, and then to abstraction, where the interventions of interest are determined by the map from low-level states to high-level states, and strong abstraction, which takes more seriously all potential interventions in a model, not just the allowed interventions. We show that procedures for combining micro-variables into macro-variables are instances of our notion of strong abstraction, as are all the examples considered by Rubenstein et al.
1 Introduction
The paper asks when a high-level causal model faithfully abstracts a low-level model, especially when micro-variables are clustered into macro-variables. It critiques exact transformation for allowing overly broad inessential differences and develops stricter abstraction notions.
- Motivation: Causal abstraction asks whether a high-level model faithfully represents causal relationships in a low-level model.The motivation spans analyses from neurons to beliefs and from individual voters to social groups.
- Motivation: Clustering micro-variables into macro-variables must preserve causal relationships, since equal aggregate values can arise from settings with different outcomes.The paper illustrates this with X + Y + Z, where distinct settings may produce the same sum but different effects.
- Prior work: Rubenstein et al. define exact transformation between causal models and interpret the target model as an abstraction of the source model.Their framework is the paper’s starting point for analyzing abstraction.
- Problem: The paper argues that exact transformation permits an overly broad notion of which differences between models are inessential.It frames the issue as application-dependent but maintains that the definition masks differences that should matter.
- Contributions: The paper introduces successively more restrictive notions, culminating in constructive abstraction, where low-level variables are partitioned into cells mapped to unique high-level variables.It presents constructive abstraction as the special case most directly matching procedures that combine micro-variables into macro-variables.
- Contributions: Constructive abstraction is presented as likely most useful in practice, while abstraction and strong abstraction remain theoretically interesting.The paper also aims to capture related abstraction intuitions while emphasizing additional causal subtleties.
2 Probabilistic causal models: a review
This section reviews causal models as structural-equation systems with endogenous and exogenous variables, contexts, interventions, and optionally probabilities over contexts. It also states the recursive-model and unique-exogenous-variable assumptions used in the paper.
- Causal-model components: A causal model consists of a signature, structural equations, and a set of allowed interventions.The signature specifies exogenous variables, endogenous variables, and each variable’s possible values.
- Causal-model components: Structural equations determine endogenous variables from other endogenous and exogenous variables, while exogenous assignments form contexts.No structural functions are associated with exogenous variables because their values are determined outside the model.
- Model assumptions: Recursive models impose an acyclic dependency order, allowing variable values to be determined sequentially for each context.In strongly recursive models, this order is independent of context, and equations can be written using only the variables on which each variable depends.
- Interventions: An intervention sets selected endogenous variables to specified values by replacing their structural equations with constant assignments.All equations for variables outside the intervention remain unchanged.
- Scope and assumptions: The paper considers allowed interventions and restricts its exposition to recursive models, while noting that its definitions and results extend to a more general unique-solution setting.The unique-exogenous-variable assumption is not the source of the problems identified with the earlier abstraction notions.
- Causal formulas: Causal formulas describe what would hold under interventions, with truth evaluated in a model and context.The notation [Y1 ← y1, ..., Yk ← yk]ϕ expresses that ϕ holds when the selected variables are set to those values.
- Probabilistic models: A probabilistic causal model adds a probability distribution over contexts to a causal model.The paper reviews equivalence as agreement on probabilities of all causal formulas, and states that every probabilistic model has an equivalent model with unique exogenous variables.
3 From exact transformations to abstractions
The paper progressively strengthens causal-model abstraction to prevent probability choices or intervention restrictions from masking substantive differences. It moves from exact and uniform transformations to abstraction and strong abstraction, clarifying how state mappings determine interventions and how broader intervention sets expose failures.
- Exact transformations: Exact transformations can be satisfied by choosing suitable low- and high-level distributions, even when the high-level model is not a meaningful abstraction.The paper identifies this flexibility as a central problem with the RW+ definition.
- Uniform transformations: Uniform transformations also induce a context mapping that determines the corresponding high-level distribution as the pushforward of the low-level distribution.For countable intervention sets, Proposition 3.6 guarantees a function τU producing the high-level distribution from every low-level distribution.
- Abstraction: Abstraction makes the intervention mapping depend on the state map τ and does not require a specified probability distribution.This distinguishes τ-abstraction from exact transformation while preserving the implication from abstraction to uniform transformation.
- Strong abstraction: Strong abstraction considers the largest possible intervention sets, revealing cases where high-level interventions have no meaningful low-level counterpart or where uniformity fails.The energy example fails under interventions fixing mass, although restricting low-level interventions yields a useful weaker abstraction.
- Strong abstraction: The intervention sets induced by τ can exclude logically impossible high-level interventions, as illustrated by the constrained pixel-grid example.Some high-level interventions cannot equal ωτ of any low-level intervention because the allowed high-level states obey a relation between their components.
H. It is straightforward to check that (MH, Iτ
The paper distinguishes strong abstraction from constructive abstraction and shows how intervention restrictions can prevent strong abstraction, while micro-variable clustering motivates the constructive form.
- Strong versus constructive abstraction: A strong abstraction may fail when high-level variables can only be intervened on simultaneously, limiting separate interventions on those variables.The paper contrasts this with a model using one combined variable, which can satisfy strong abstraction.
- Strong versus constructive abstraction: A second example shows strong abstraction can fail because the empty intervention and X3 ←0 map to the same high-level intervention while inducing incompatible probabilities.The contradiction requires c = 1, so the construction fails when c < 1.
- Strong versus constructive abstraction: In one example, allowing only low-level interventions that set X3 to 0 yields a τ-abstraction but not a strong τ-abstraction.The corresponding intervention set is restricted, with Iτ2 = I*2.
- From micro-variables to macro-variables: Constructive abstraction partitions low-level variables into groups mapped separately to high-level variables, with any remaining variables marginalized away.Each nonempty group corresponds to one macro-variable, while the final group may be empty.
- From micro-variables to macro-variables: Every constructive τ-abstraction is strong, but the converse is only conjectured under additional technical conditions.The paper has not proved this converse.
- From micro-variables to macro-variables: The authors expect constructive abstractions to arise most often in practice and report that all three Rubenstein et al. examples are constructive abstractions.They also state that one additional example can be extended to become constructive by adding interventions.
4 Discussion and Conclusions
The paper presents abstraction as a formal way to relate causal models at different detail levels while preserving faithful high-level reasoning. It positions the framework as groundwork for approximate abstraction, cross-model actual causation, and abstraction of complex models.
- Discussion and Conclusions: The framework distinguishes abstraction notions by the kinds of causal models and intervention relations they connect.τ-abstraction relates basic causal models, uniform transformation relates causal models, and exact transformation relates probabilistic causal models.
- Discussion and Conclusions: The authors view a good abstraction notion as critical for reasoning at a high level while remaining faithful to a more detailed causal model.The paper frames this as a central motivation for formalizing relations between levels of detail.
- Discussion and Conclusions: The framework is intended as formal groundwork for extending abstraction to approximately correct mappings in realistic settings.The authors identify approximate abstraction as a direction for future work.
- Discussion and Conclusions: Future applications include studying actual causation across causal models and abstracting highly complex causal models into simpler ones.The paper gives as an example an event in a low-level model causing an event in a high-level model.
A Appendix: Proofs
The appendix proves correspondences between uniform transformations and exact transformations, constructs compatible exogenous mappings, and establishes order preservation for intervention mappings.
- Model construction: A constructed model replaces shared exogenous dependence with one exogenous variable per endogenous variable while preserving the original model’s behavior.The new distribution assigns probability only to tuples whose component contexts are identical, reproducing the original context distribution.
- Model construction: The construction defines each new structural equation to depend only on its corresponding exogenous variable, thereby giving the model unique exogenous variables.The resulting model is shown equivalent to the original model under the constructed probability distribution.
- Uniform transformation correspondence: For a uniform transformation, low-level and high-level contexts correspond when mapped low-level outcomes equal high-level outcomes under every allowed intervention.This correspondence supports constructing a map from low-level contexts to high-level contexts.
- Uniform transformation correspondence: The proof constructs τU by selecting a corresponding high-level context for each low-level context and then maps low-level distributions through τU.The resulting transformed model satisfies the exact transformation condition.
- Exact transformation equality: The probability calculation derives the high-level outcome probability from the mapped low-level outcomes and the state map τ.This establishes the required equality for exact transformation.
- Intervention ordering: The intervention map is order-preserving: refining a low-level intervention produces a high-level intervention that is at least as refined.The proof uses inclusion of reachable-state sets under τ to establish the ordering.