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arxiv: 2605.22866 · v1 · pith:C5POULQLnew · submitted 2026-05-19 · 💻 cs.AI · cs.LG

BOHM: Zero-Cost Hierarchical Attribution for Compound AI Systems

Joss Armstrong This is my paper

Pith reviewed 2026-05-25 05:50 UTC · model grok-4.3

classification 💻 cs.AI cs.LG
keywords hierarchical attributionrouting weightscompound AI systemszero-cost attributionmulti-resolution decompositionShapley comparisonagentic systemsorchestrator routing
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The pith

BOHM derives multi-resolution attributions for compound AI systems directly from the routing weights the orchestrator already maintains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Compound AI systems route tasks through hierarchies of specialised components, yet Shapley-based attribution requires evaluating arbitrary coalitions of those components, which is often impossible for third-party APIs or when routers concentrate on a few tools. BOHM instead extracts a full attribution tree from the existing routing weights: each leaf receives the product of weights along its root-to-leaf path, and each level-k node receives the induced distribution obtained by summing the path products that reach it. The construction runs at zero marginal cost, needs no internal access to components, and supplies attributions at every depth simultaneously. On LLM hierarchies, agentic driver studies, and a census data tree the resulting rankings correlate strongly with SHAP when the router selects near-optimal tools and recover ground-truth orderings at multiple resolutions. BOHM satisfies efficiency, monotonicity, symmetry and weak suppression but not additivity, positioning it as a complementary primitive rather than a replacement for coalition methods.

Core claim

BOHM defines leaf attribution as the product of routing weights along the path from root to leaf and level-k attribution as the probability distribution over depth-k nodes induced by those same path products. The tree is obtained directly from the routing state already kept by the orchestrator, requires no additional system evaluations or component introspection, and yields simultaneous attributions at every level of the hierarchy.

What carries the argument

The hierarchical attribution tree whose leaves are path products of routing weights and whose internal nodes are the summed products reaching each depth-k node.

If this is right

  • Attributions become available at every depth of the hierarchy in a single pass without extra coalition evaluations.
  • Opaque or third-party components receive attributions because the method never inspects their internals.
  • Disagreement between BOHM and SHAP on a given cell indicates that the deployed router is not selecting the empirically strongest tool.
  • The method satisfies efficiency, monotonicity, symmetry and weak suppression while remaining computable wherever routing state exists.
  • Multi-resolution attribution is obtained wherever a router already maintains path probabilities, regardless of the evaluation budget available for flat methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • BOHM could function as an immediate, zero-cost diagnostic for whether an orchestrator's routing policy is near-optimal on the tasks it actually encounters.
  • The same path-product construction might apply to any tree-structured decision process that records routing probabilities, including non-AI planning or decision trees.
  • When BOHM and SHAP diverge, the difference itself supplies information about the gap between the router's policy and the true contribution surface.
  • Extending the method to routers whose weights change over time would require checking whether the path products remain stable or need re-aggregation.

Load-bearing premise

That the routing weights maintained by the orchestrator can be interpreted as multiplicative factors whose path products yield meaningful per-component attributions.

What would settle it

An experiment that independently measures each component's true marginal contribution on a held-out set of tasks, then checks whether the ordering produced by path-product attributions matches the measured ordering when the router is forced to use a demonstrably suboptimal tool.

Figures

Figures reproduced from arXiv: 2605.22866 by Joss Armstrong.

Figure 1
Figure 1. Figure 1: BOHM attribution on 18 LLMs in a 3-level hierarchy over 880 LiveCodeBench problems. [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: BOHM vs SHAP attribution on 18 LLMs over 880 LiveCodeBench problems. Left: [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Structure sensitivity: natural (quality-based) vs random grouping. Left: Kendall [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Opaque-component attribution. 9 components with hidden quality parameters. BOHM [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Per-level attribution trajectories. The root’s weights reflect group-level attribution; each [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Attribution quality as a function of hierarchy depth (branching factor 3) and number [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Attribution quality vs outcome noise (probability of flipped binary outcome). Graceful [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Context-dependent attribution. Different contexts (text, code, math) activate different [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Domain-conditioned attribution across five coding benchmarks. Top row: per-model [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Fixed (LCB-based) vs domain-specific tiering. The HumanEval delta ( [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
read the original abstract

Compound AI systems route tasks through hierarchies of specialised components. Attribution is dominated by Shapley-based methods (SHAP), which decompose a coalition value function into per-component marginal contributions and require evaluation of the system on arbitrary component subsets. 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. We introduce BOHM, which extracts a hierarchical attribution tree directly from the routing weights such systems already maintain: leaf attribution is the path product of root-to-leaf routing weights; level-k attribution is the induced distribution over depth-k nodes. 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. BOHM and SHAP answer different questions and converge when the deployed router routes near-optimally. On 18 LLMs in a 3-level hierarchy over 880 LiveCodeBench problems, BOHM yields Kendall tau=0.928; SHAP reaches tau=0.980 at 9,000x more coalition evaluations per seed. On a 5-driver, 7-benchmark agentic study (35 cells, complete coverage), drivers concentrate routing on a single tool (top-share median 0.65), and cell-level tau(BOHM,SHAP) is predicted by whether the driver's top pick is the empirically best tool (mean +0.22 vs ~+0.01). On a US Census hierarchy (475 leaves, 4 levels), BOHM recovers ground-truth rankings at every level (tau up to 0.722). BOHM satisfies efficiency, monotonicity, symmetry, and weak suppression but not Shapley's additivity. It is best understood as a complementary primitive: a multi-resolution decomposition computable wherever routing state exists, whose disagreement with Shapley is itself diagnostic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces BOHM, a zero-cost hierarchical attribution method for compound AI systems. It defines leaf attribution as the product of routing weights along each root-to-leaf path and level-k attribution as the induced marginal distribution over depth-k nodes. The method requires no component internals or additional evaluations, provides multi-resolution output simultaneously, satisfies efficiency/monotonicity/symmetry/weak suppression (but not additivity), and reports Kendall-tau agreement with SHAP of 0.928 on 18 LLMs over LiveCodeBench, up to 0.722 on Census data, and a link between agreement and routing quality in an agentic study.

Significance. If the reported correlations and diagnostic property hold, BOHM supplies a practical complementary primitive for attribution in opaque, third-party, or agentic systems where coalition-based methods like SHAP cannot be applied at scale. The zero marginal cost, hierarchical structure, and explicit axiom verification are clear strengths that flat attribution methods lack at any budget.

major comments (2)
  1. [Abstract] Abstract: the reported Kendall-tau agreement (0.928 on LLMs, 0.722 on Census) is presented without error bars, number of seeds, or explicit data-exclusion rules; these omissions are load-bearing for the claim that agreement is high and diagnostic of routing quality.
  2. [Abstract] Method definition (Abstract and results): leaf attribution is defined directly as the path product of routing weights, yet no derivation, approximation guarantee, or error bound is supplied showing why this quantity equals or approximates marginal contribution to system output—the quantity targeted by the SHAP comparisons; support remains conditional on near-optimal routing regimes without a general test or counter-example family.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'SHAP reaches tau=0.980 at 9,000x more coalition evaluations per seed' would benefit from a brief parenthetical on how the 9,000x factor was computed (e.g., exact coalition count per instance).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and outline revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the reported Kendall-tau agreement (0.928 on LLMs, 0.722 on Census) is presented without error bars, number of seeds, or explicit data-exclusion rules; these omissions are load-bearing for the claim that agreement is high and diagnostic of routing quality.

    Authors: We agree that the abstract would benefit from additional statistical context. The full experimental section details the 880 LiveCodeBench problems across 18 LLMs and the 475-leaf Census hierarchy, along with the agentic study's 35 cells. We will revise the abstract to reference the number of seeds (where multiple runs were performed) and add a parenthetical note on variability or direct the reader to the experimental details. Data-exclusion rules (e.g., complete coverage in the agentic study) will be stated explicitly in a revised abstract or methods footnote. revision: yes

  2. Referee: [Abstract] Method definition (Abstract and results): leaf attribution is defined directly as the path product of routing weights, yet no derivation, approximation guarantee, or error bound is supplied showing why this quantity equals or approximates marginal contribution to system output—the quantity targeted by the SHAP comparisons; support remains conditional on near-optimal routing regimes without a general test or counter-example family.

    Authors: The manuscript positions BOHM as extracting attributions directly from existing routing weights rather than approximating marginal contributions in general; it explicitly states that the two methods answer different questions and converge when routing is near-optimal. The agentic study (35 cells) provides an empirical test linking agreement to routing quality (mean +0.22 when top pick matches best tool). We will add a short methods paragraph deriving the path-product definition from the routing probability interpretation and clarifying the conditional nature of agreement, without claiming a general approximation guarantee. A counter-example family is not required because disagreement is presented as a diagnostic feature rather than a flaw. revision: yes

Circularity Check

0 steps flagged

No significant circularity; BOHM is a direct definition from routing weights with external empirical checks

full rationale

The paper defines leaf attribution explicitly as the path product of routing weights and level-k attribution as the induced marginal distribution; these are introduced as a construction rather than derived from first principles as equivalent to marginal contributions. It states that BOHM and SHAP answer different questions and converge only when the router is near-optimal, with Kendall-tau agreement reported as an external measurement on benchmarks. No self-citations appear in the load-bearing steps, no parameters are fitted to target attributions, and the axiom checks (efficiency, monotonicity, etc.) are performed directly on the defined quantities. The derivation chain is therefore self-contained as a zero-cost extraction method whose relation to Shapley values is measured rather than assumed by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method introduces no free parameters or invented entities. It rests on one domain assumption about the semantic meaning of routing weights.

axioms (1)
  • domain assumption Routing weights maintained by the system can be interpreted as selection factors whose path products yield meaningful attributions.
    Invoked when defining leaf attribution as the path product and level-k distributions.

pith-pipeline@v0.9.0 · 5878 in / 1266 out tokens · 42366 ms · 2026-05-25T05:50:40.726358+00:00 · methodology

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Reference graph

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