Lab Note: Attempt #017

Unified Information Geometry for Causal Inference

Status: Failed Duration: 8 weeks (Oct-Dec 2025) Team: Paul Luong, 2 collaborators

Context

Causal inference has multiple approaches: Pearl's graphical models, potential outcomes framework, information-theoretic methods. Each has strengths but they remain conceptually fragmented. We hypothesized that information geometry could provide a unified foundation.

Motivation

Information geometry successfully unified statistics and information theory
Causal relationships have clear information-theoretic interpretations
A geometric framework would enable new inference algorithms
Unification would clarify relationships between existing methods

Hypothesis

Primary Claim: There exists a Riemannian manifold structure where:

Points represent causal models
Geodesics correspond to minimal interventions
Different inference frameworks are coordinate systems on the same manifold

Expected Benefit: Unified theory enabling translation between frameworks and new algorithmic approaches.

Approach

Phase 1: Theoretical Development

We developed a geometric framework:

M : CausalManifold where
  Points := CausalModels
  Metric := InformationGeometric
  Curves := InterventionPaths

Initial results were promising:

Successfully embedded simple causal models as manifold points
Derived geodesic equations for intervention design
Showed Pearl's do-calculus as special case of geometric transport

Phase 2: Extension to Complex Models

Attempted to extend framework to:

Models with latent variables
Continuous variables
Non-parametric distributions
Cyclic causal structures

This is where problems emerged.

Failure Mode

Problem 1: Incompatible Independence Assumptions

Different causal inference frameworks make incompatible independence assumptions:

Pearl's framework: d-separation on DAGs
Potential outcomes: ignorability assumptions
Information-theoretic: specific factorization properties

These cannot be simultaneously satisfied on a single geometric structure. The manifold would need different topologies for different frameworks.

Problem 2: Geodesic Non-Uniqueness

For complex models, our "minimal intervention" geodesics were non-unique. Multiple incomparable intervention paths existed. This violated the manifold structure requirement.

Problem 3: Computational Intractability

Even when the framework applied, computing the metric tensor for realistic models was intractable. The geometry requires O(2^n) calculations for n variables.

Breaking Point

The decisive failure: We attempted to represent a simple confounded model (X ← Z → Y, X → Y) in the framework. The geometry admitted contradictory geodesics depending on which independence assumption we enforced.

This wasn't a bug—it revealed a fundamental incompatibility. The different inference frameworks aren't merely different coordinate systems on the same structure. They're based on inequivalent foundations.

Extracted Insights

Insight 1: Framework Pluralism is Fundamental

The fragmentation of causal inference isn't accidental or fixable through better formalism. Different frameworks make incompatible ontological commitments about causality.

Implication: Future work should focus on translation protocols between frameworks rather than unification.

Insight 2: Geometric Methods Have Limited Scope

Information geometry works when:

Independence structure is compatible across representations
Distance metrics have unique natural definitions
Computational complexity is manageable

For causal inference, these conditions rarely hold simultaneously.

Implication: Geometric methods may work for restricted causal model classes but not generally.

Insight 3: Computational Constraints Shape Theory

The O(2^n) complexity isn't incidental—it reflects fundamental computational limits of causal reasoning. Any complete theory must grapple with this.

Implication: Practical causal inference requires approximations. Theory should account for this from the start.

What Worked

Despite overall failure, some components proved valuable:

Intervention Distance Metric: Our geometric distance between causal models via intervention is useful even without full manifold structure.
Local Geometric Structure: The framework works locally (small neighborhoods of model space). This may have applications in sensitivity analysis.
Computational Tools: The code for geometric calculations is being repurposed for other projects.

Next Directions

Alternative Approach 1: Category-Theoretic Framework

Instead of geometric unification, explore categorical relationships between frameworks. Functors between framework categories may capture translations without forcing unification.

Alternative Approach 2: Restricted Model Classes

The geometric framework may work for specific causal model classes (e.g., linear Gaussian). Investigate which restrictions enable geometric treatment.

Alternative Approach 3: Computational Approximations

Develop approximate geometric methods that sacrifice theoretical purity for tractability. The full geometry may be intractable but approximations could be useful.

Lessons for Research Process

What We'd Do Differently

Earlier Implementation: We developed too much theory before testing on realistic examples. Earlier prototyping would have revealed issues sooner.
Boundary Condition Testing: Should have systematically checked edge cases and compatibility conditions before building elaborate superstructure.
Literature Review Depth: Some incompatibility results existed in scattered literature. Deeper review would have saved time.

What We Did Right

Clear Hypotheses: We stated clear, testable claims. This made failure unambiguous.
Documentation: We documented as we worked. Reconstructing this lab note was straightforward.
Code Preservation: All code is version-controlled and annotated. Future work can build on working components.

Conclusion

Attempt #017 failed to achieve its primary goal. The dream of geometric unification for causal inference appears fundamentally flawed, at least in the form we pursued.

However, the attempt generated value:

Clarified fundamental limitations
Produced reusable components
Suggested alternative research directions
Saved time for others pursuing similar approaches

Failed research that's well-documented is still progress.

Code & Materials

Implementation available at: [internal repository] Mathematical notes: [internal documentation] Weekly logs: [internal wiki]

Related Work

Pearl's do-calculus remains the most practical framework for our purposes
Potential outcomes approach better for specific experimental designs
Information-theoretic methods valuable for communication-limited settings

We're adopting a pragmatic multi-framework approach rather than seeking unification.

Status: Project archived Extracted components: Intervention distance metric, local geometric tools Follow-up: Category-theoretic investigation (preliminary)

Contact: For technical details or related research, contact@theorome.org