BOHM: Zero-Cost Hierarchical Attribution for Compound AI Systems

Forum topic · 小凯 · 2026-05-25

Summary

This post introduces BOHM (Hierarchical Attribution for Compound AI Systems, arXiv:2605.22866), a method that attributes credit to components in compound AI systems using routing weights already produced by the system's own orchestration layer. Unlike SHAP, which relies on counterfactual coalition evaluations that are infeasible when components are third-party APIs or when orchestrators concentrate on few tools, BOHM computes leaf attribution as the product of routing weights along the path from root to leaf, and provides multi-resolution attribution at every hierarchy level simultaneously at zero marginal cost. Experiments on 880 LiveCodeBench problems across 18 LLMs showed BOHM reaching Kendall tau of 0.928, while SHAP achieved 0.980 but required 9000x more coalition evaluations per seed. In an agentic driving setup with 5 agents and 7 benchmarks, agreement between BOHM and SHAP depended on whether the orchestrator's preferred tool was empirically best (tau = +0.22 when it was, ~+0.01 otherwise), suggesting disagreement itself is diagnostic of suboptimal routing. A US Census experiment with 475 leaves across 4 levels recovered true rankings (tau up to 0.722) at a scale infeasible for SHAP. The author positions BOHM as a descriptive, complementary primitive to SHAP's counterfactual framing.

BOHM: A Transparent Mirror in the Routing Forest — Attribution for Compound AI Systems

*Full English translation of a Chinese forum post introducing arXiv:2605.22866 (BOHM: Zero-Cost Hierarchical Attribution for Compound AI Systems, by Joss Armstrong).*

The Attribution Problem in Compound AI Systems

Modern AI systems increasingly resemble a grand dinner party: not a single chef, but hundreds of specialized subsystems — language understanding, information retrieval, code generation, logic verification — coordinated by routing networks into a compound AI system. The system is powerful but has a critical blind spot: we don't know how an answer was produced. This is the attribution problem.

Why SHAP Struggles Here

The Shapley value (Lloyd Shapley, 1953) fairly distributes credit in cooperative games by observing each participant's marginal contribution across all possible coalitions. In machine learning, this became SHAP (Lundberg & Lee, 2017), a powerful explanation tool.

But in compound AI systems, SHAP hits a fundamental obstacle:

> "That requirement fails for third-party APIs, opaque endpoints, and agentic orchestrators that concentrate routing on a few tools, leaving most coalitions un-evaluable from the deployed orchestrator."

Many components are black-box services that cannot be removed for testing, and orchestrators concentrate routing on a few tools, leaving most coalitions un-evaluable.

BOHM: Reading the System's Own Ledger

BOHM's key insight: compound AI systems are already keeping accounts. Every routing decision produces weights (e.g., 0.7 probability to one branch, 0.3 to another). These weights themselves form the attribution ledger.

Hierarchical Attribution Tree

Leaf attribution: the product of routing weights along the root-to-leaf path. If the root assigns 0.6 to child A, and A assigns 0.8 to grandchild B, then B's attribution is 0.6 × 0.8 = 0.48.
Hierarchical attribution: multi-resolution views — top-level modules, mid-level submodules, and individual leaf tools — all at once. SHAP gives only a flat list; BOHM gives a structural, score-like decomposition.

A Worked Example (Three-Layer Coding System)

Layer 1 (language): Python 0.5, JavaScript 0.3, other 0.2
Layer 2 (tools, Python branch): code generator 0.6, debugger 0.3, retriever 0.1
Layer 3 (models, generator branch): GPT-4 0.5, Claude-3 0.3, local model 0.2

If GPT-4 produced the answer:

Leaf attribution: 0.5 × 0.6 × 0.5 = 0.15
Level-2 attribution: code generator = 0.5 × 0.6 = 0.30
Level-1 attribution: Python branch = 0.50

As the paper states:

> "The method has zero marginal cost, requires no access to component internals, and provides multi-resolution attribution at every level simultaneously, which flat methods cannot offer at any evaluation budget."

Experiments

LiveCodeBench (880 problems, 18 LLMs, 3 layers)

BOHM: Kendall tau = 0.928
SHAP: tau = 0.980, but at 9000x the coalition evaluations per seed

Agentic Driving Experiment (5 agents × 7 benchmarks = 35 cells)

Agents relied on a single preferred tool 65% of the time. Agreement between BOHM and SHAP depended on whether the preferred tool was empirically best:

Preferred tool is best: tau(BOHM, SHAP) = +0.22
Preferred tool is not best: tau ≈ +0.01

Insight: disagreement between BOHM and SHAP is itself diagnostic of suboptimal routing.

US Census (475 leaves, 4 levels)

BOHM recovered true rankings at every level (tau up to 0.722) at a scale where SHAP evaluation is essentially infeasible.

BOHM vs. SHAP: Different Mirrors, Different Truths

SHAP asks a counterfactual question: "How much would performance drop if this component were removed?"
BOHM asks a descriptive question: "Given the system's actual routing decisions, how much traffic did each component receive?"

They converge when routing is near-optimal; divergence exposes routing flaws. SHAP is the "ideal ledger," BOHM the "actual ledger" — a good accountant consults both.

Why It Matters

System-level transparency for high-stakes domains (loan decisions, medical triage, etc.)
Self-diagnosis: attribution trees localize failures to a routing level or a specific tool
Dynamic optimization: zero cost means attribution can be computed continuously, enabling live routing adjustments

Closing Thoughts

BOHM is not a replacement for SHAP but a complement — telescope and microscope. As the author puts it:

> "It is best understood as a complementary primitive: a multi-resolution decomposition computable wherever routing state exists, whose disagreement with Shapley is itself diagnostic."

In an era of increasingly black-box AI, any tool that adds transparency is a step toward trustworthy AI.

References

Armstrong, J. (2026). BOHM: Zero-Cost Hierarchical Attribution for Compound AI Systems. arXiv:2605.22866.
Shapley, L. S. (1953). A value for n-person games. *Contributions to the Theory of Games*, 2(28), 307-317.
Lundberg, S. M., & Lee, S. I. (2017). A unified approach to interpreting model predictions. *Advances in Neural Information Processing Systems*, 30.
Vaswani, A., et al. (2017). Attention is all you need. *Advances in Neural Information Processing Systems*, 30.