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When Agents Disagree: Bayesian Backward Reasoning as a Label-Free Anchor for Multi-Agent Collective Decision-Making

Ken Chen, Wei Wang, Sachith Seneviratne, Hansani Weeratunge, Saman Halgamuge

arXiv:2609.11709v1cs.AIcs.MA

TL;DR

When agents disagree, voting and LLM judges remain on the forward evidence-to-conclusion path and can inherit correlated errors. This paper constructs a reverse posterior and uses cross-path consistency to guide selection, reweighting, and fusion; on DDXPlus across five backbones, the strategies improve over baselines, with the largest gains on Disagree.

  • Problem

    Voting and LLM judges rely on forward reasoning and can inherit correlated errors within the agent pool, motivating a differently factorized reference for conflicting answers.

    Source: §1 · p. 1, §6 · p. 9

  • Method

    The paper constructs a shared reverse posterior through Bayesian backward reasoning and uses its Jensen-Shannon consistency with forward posteriors to guide three training-free aggregation strategies.

    Source: §1 · p. 2, §6 · p. 9

  • Results

    Across five LLM backbones on DDXPlus, reverse-anchored aggregation improves over baselines, with the largest gains on Disagree; LogLin strictly outperforms all baselines across scenarios.

    Source: §1 · p. 2, §6 · p. 9

  • Takeaways & Limitations

    Although often weaker as a standalone predictor, the reverse posterior provides useful complementary anchor information, and replacing it with MeanF or GenF degrades FwdJS and LogLin.

    Source: §6 · p. 9

    Evaluation is limited to single-round, same-backbone pools on the synthetic closed-set DDXPlus benchmark, so mixed-backbone, multi-round, and other evidence-to-label settings remain untested.

    Source: Limitations · p. 10

Abstract

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When multiple LLM agents yield conflicting answers, the decision-making process dictates whether agent diversity improves performance or merely compounds shared errors. Existing collective decision-making methods, including voting, electoral rules, and LLM judges, rely on forward reasoning: they map evidence to labels in one direction. Although these methods can combine diverse forward traces, they still aggregate estimates that share this evidence-to-label factorization and can inherit correlated errors within the forward pool. We therefore construct a reverse posterior for each instance through Bayesian backward reasoning from an explicit likelihood. The forward and reverse posteriors provide differently factorized approximations of the underlying posterior. Because estimates from different factorizations may tend to share the same error less often, we use Jensen-Shannon divergence to rank agents by cross-path consistency. This cross-path consistency signal underlies three strategies: hard selection (MinJS), soft reweighting (FwdJS), and log-linear fusion (LogLin). Evaluated on DDXPlus across five LLM backbones, our proposed strategies show consistent improvements: MinJS outperforms random selection across all backbones, FwdJS generally improves over the strongest baseline, and LogLin achieves the best performance among the evaluated methods, with its largest gains on the subset where the agents disagree. Despite its weaker standalone accuracy, the reverse posterior serves as a more useful anchor than forward-only alternatives, providing complementary information for collective decision-making. When labeled data are available, a lightweight two-stage calibration can further refine the reverse anchor and improve aggregation performance.

1 Introduction

When agents disagree, forward-only voting and judging can inherit correlated errors and lack a reliable external anchor. The paper constructs a reverse posterior and uses its cross-path consistency with forward posteriors to guide three training-free aggregation strategies.

1 Introduction

  • Voting, electoral rules, and LLM judges rely on the same forward reasoning path as agents, so correlated errors can persist when agents disagree.The introduction identifies a differently factorized reference as the missing anchor for resolving conflicting answers.Source: §1 · pp. 1–2
  • Bayesian backward reasoning constructs a shared reverse posterior from an explicit likelihood and prior, providing a differently factorized approximation of the class posterior.The forward and reverse posteriors are treated as biased estimates whose errors may be less collinear across paths.Source: §1 · p. 2
  • Jensen-Shannon divergence between each forward posterior and the reverse posterior ranks agents by cross-path consistency, not as a certificate of correctness.The paper argues that distance from internal consensus can reward conformity, including when the majority is wrong.Source: §1 · p. 2
  • MinJS selects the forward agent closest to the reverse posterior, while FwdJS uses that posterior to set weights without contributing probability mass.Hard selection can be brittle because compatibility with the reverse posterior does not guarantee correctness.Source: §1 · p. 2
  • LogLin adds the reverse posterior as a multiplicative factor with fixed weight wR=0.2, combining cross-path information absent from the forward-only strategies.The contribution suite reports MinJS outperforming the selection baseline, FwdJS improving over the strongest baseline in most settings, and LogLin strictly outperforming baselines across scenarios.Source: §1 · p. 2
  • Replacing the reverse anchor with MeanF or GenF strictly degrades FwdJS and LogLin, while two-stage calibration can refine the anchor when limited labeled data are available.The paper describes the reverse posterior as useful despite often being the weakest standalone predictor.Source: §1 · p. 3

2 Related Work

Prior work aggregates forward agent outputs through voting or judging, uses backward reasoning mainly for candidate verification, or fuses forward predictions for ensembling and routing. This paper distinguishes its approach by using a full reverse posterior as a reference for collective decisions.

Multi-agent collective decision-making

  • Voting, electoral rules, and designated judges make collective decisions from agents’ forward outputs, while the paper derives an instance-level reference from a generative reverse posterior.Related work also describes confidence-weighted consensus and analyses comparing majority voting with multi-round debate.Source: §2 · p. 3

Forward–backward reasoning

  • Self-Verification, FOBAR, and RevThink use backward reasoning primarily to verify candidate-level predictions, whereas this work uses a full reverse posterior to guide multi-agent aggregation.The reverse signal guides selection, reweighting, and fusion across agents.Source: §2 · p. 3

Ensembling and unlabeled routing

  • Ensembling and unlabeled routing combine forward predictions or score models, while this work retains explicit agents and uses a reverse reference rather than another forward agent or learned judge.The related-work passage names LLM-Blender, DeePEn, PackLLM, and SMOOTHIE as examples of these alternatives.Source: §2 · p. 3

3 Method

The method uses a shared reverse posterior as a consistency reference for collective decisions over forward-agent posteriors. It supports hard selection, soft reweighting, forward–reverse fusion, and optional labeled calibration.

3.1 Collective decision over agent posteriors

  • Collective decisions select one agent posterior or fuse the pool into a single posterior, with both forms implemented in a label-free, training-free inference framework.The fusion output is classified by its maximum-posterior label.Source: §3.1 · p. 3

3.2 Reverse posterior as a consistency anchor

  • The shared reverse posterior R is constructed from contextual evidence a and remaining evidence e under a→d→e, using P(e | d) and P(d | a).The method treats forward and reverse inference as differently conditioned perspectives on the same latent label, not as exact posterior estimates.Source: Figure 1 · p. 4, §3.2 · p. 4

3.3 Reverse-referenced selection and fusion

  • MinJS selects the agent whose posterior is most consistent with the reverse anchor; FwdJS softly reweights forward posteriors, and LogLin additionally incorporates the reverse posterior.MinJS uses the anchor only to rank agents, while FwdJS uses it to set weights; both outputs use forward posteriors, unlike LogLin.Source: §3.3 · p. 5

3.4 Labeled calibration of the reverse anchor

  • With labeled training data, calibration adjusts the ordinal likelihood and contextual-activation maps, rescales R, then applies a scalar class-prior correction.The calibrated maps and temperature are frozen before recomputing R; the prior correction uses a single scalar rather than class-specific bias.Source: §3.4 · pp. 5–6

3.5 Computational cost

  • Inference uses K forward-agent inferences plus one shared reverse inversion per instance, then divergence computation and aggregation, with no test-time parameter updates.The reverse construction is shared across aggregation operators; labeled calibration, if used, is fitted only on a training split.Source: §3.5 · p. 6

4 Experimental Setup

Experiments use a fixed DDXPlus test slice and compare reverse-referenced aggregation with forward-only selection, voting, and judge baselines. Five heterogeneous agents are evaluated on each of five LLM backbones.

4.1 Dataset

  • DDXPlus provides structured clinical evidence and 49 disease labels; evaluation uses a fixed 2,000-case test slice and reports All and the non-unanimous Disagree subset.The per-backbone evaluation pool is the intersection of complete forward cases and cases with an available reverse posterior, held fixed across methods.Source: §4.1 · p. 6

4.2 Baselines

  • Baselines include random selection, voting rules over forward ballots or posteriors, and two LLM judges over the same five agents.Random Agent uniformly samples a forward agent and serves as MinJS’s selection control.Source: §4.2 · p. 6

4.3 Models and implementation details

  • Five prompting strategies produce top-5 posteriors over 49 labels for each of five backbones, using temperature 0.6, top-p 0.95, and one response per agent per query.The primary metric is Gold Top-Pathology Accuracy at 1 (GTPA@1); FwdJS uses τ = 5.0 and LogLin uses wR = 0.2.Source: §4.3 · p. 6

5 Results

LogLin leads the reported collective-decision comparisons across backbones, while the reverse anchor’s utility is not predicted by its standalone accuracy. Reverse errors collide less often with forward consensus errors, and JS distance to the reverse anchor provides a more consistent agent ranking than forward-only anchors; calibration further improves aggregation.

5.1 Main Results: Reverse-Anchored Collective Decision

  • On Disagree, LogLin beats the strongest forward-only electoral rule by 1.2–4.7 pp across five backbones and achieves the best result for every backbone on All.The reported Disagree gains are diluted in the All aggregate, which also includes cases where forward agents agree.Source: §5.1 · p. 7
  • MinJS beats random selection on every backbone, while FwdJS exceeds the strongest electoral rule on Disagree by 0.4–3.5 pp.Hard selection can still trail the strongest election; on Qwen All, MinJS reaches 71.64% versus 72.69% for Range voting.Source: §5.1 · p. 7

5.2 Anchor Utility Is Not Standalone Accuracy

  • On All, R is the weakest standalone predictor among R, MeanF, and GenF, trailing MeanF by 7.8–26.4 pp, yet yields the highest FwdJS and LogLin performance as anchor.Replacing R with MeanF or GenF lowers both fusion heads on every backbone.Source: §5.2 · p. 7
  • On Disagree, neither likelihood-only nor contextual-prior-only reverse anchor matches full R with LogLin on Qwen, Mistral, GLM, or DeepSeek.Llama is the exception: Rprior reaches 51.69%, slightly above R at 50.90%.Source: §5.2 · p. 7

5.3 What the Reverse Anchor Adds Beyond the Forward Pool

  • Across five backbones, plurality and R share the same incorrect label less often than plurality and either GenF or an in-pool agent: π(F, R) is 0.196–0.413.The corresponding ranges are 0.594–0.765 for π(F, GenF) and 0.680–0.829 for π(F, Fi).Source: §5.3 · pp. 7–8
  • JS(Fi, R) ranks agents more consistently by accuracy than MeanF or GenF, though R’s rank curve is not strictly monotone on every backbone.Figure 2 shows GTPA@1 by JS rank on Qwen Disagree, with R, MeanF, and GenF as anchors and a random-agent reference line.Source: §5.3 · p. 8, Figure 2 · p. 8

5.4 Calibrating the Reverse Anchor

  • On All, calibrating R improves LogLin by 0.05–2.83 pp, FwdJS by 0.20–1.51 pp, and MinJS by 1.06–4.64 pp across all five backbones.On Disagree, LogLin improves by 0.59–4.51 pp on four backbones, with Mistral the sole exception at −0.26 pp.Source: §5.4 · pp. 8–9

6 Conclusion

The paper uses a Bayesian reverse posterior as a structurally distinct reference for aggregating conflicting LLM agents. Across five backbones on DDXPlus, reverse-anchored aggregation improves over baselines, with the largest gains on disagreement cases.

6 Conclusion

  • The shared reverse posterior R supplies a structurally distinct reference, and JS(Fi, R) guides collective decision-making.The signal drives hard selection (MinJS), reverse-guided reweighting (FwdJS), and lightweight forward–reverse fusion (LogLin).Source: §6 · p. 9
  • Across five LLM backbones on DDXPlus, reverse-anchored aggregation improves over baseline methods, with the largest gains on Disagree.The reverse posterior can be a weaker standalone predictor than forward-only alternatives while serving as a more useful anchor; two-stage calibration further improves the aggregation rules.Source: §6 · p. 9

Limitations

The discussion bounds the reverse anchor's intended setting, reports how JS ranking behaves across backbones, and outlines a two-stage labeled calibration protocol.

Limitations

  • R targets finite-label evidence-to-label aggregation, while evaluation is limited to single-round, same-backbone DDXPlus and is not clinical decision support.The paper identifies mixed-backbone pools, multi-round debate, and additional evidence-to-label domains as extensions.Source: Limitations · pp. 9–10
  • The reliability level at which a weak R may begin to hurt aggregation remains uncharacterized, as does calibration sample efficiency under smaller labeled sets.The authors identify progressively smaller labeled sets as a way to determine how much supervision is sufficient for reliable anchor refinement.Source: Limitations · pp. 9–10

A JS-based agent ranking across backbones

  • On Disagree, reverse-JS rank curves are strictly monotone for Qwen and Mistral; Llama and GLM peak after rank 1, while DeepSeek's first two ranks differ by one case.The ranking is less consistently monotone under MeanF and GenF.Source: A JS-based agent ranking across… · p. 12, Table 6 · p. 12

B.1 Fitting protocol

  • Calibration fits reverse-posterior maps and temperature by label NLL, freezes them, then fits scalar γ by k-fold NLL; the appendix also details a higher-capacity bias variant.Evaluation reuses the frozen MinJS, FwdJS, and LogLin heads on the same pool as the label-free results.Source: §B.1 · p. 12, B Labeled calibration: fitting and… · p. 12

B.2 Capacity upper bound (L5CURVE_L1GB)

The higher-capacity class-specific calibration variant improves standalone reverse accuracy and downstream fusion, but is treated as an upper-bound reference because its gains may reflect label-specific priors.

B.2 Capacity upper bound (L5CURVE_L1GB)

  • Class-specific bias calibration further improves standalone R accuracy and downstream fusion, but is used only as an upper bound because gains may reflect label-specific prior adjustment rather than better routing.Table 7 reports test GTPA@1 gains over uncalibrated R with the same evaluation pool.Source: §B.2 · p. 12, Table 7 · p. 12

C Likelihood-only and prior-only reverse anchors

The paper compares the full reverse posterior with likelihood-only and prior-only reverse constructions. On Disagree, neither one-factor anchor matches full R under LogLin on four backbones; Llama is the stated exception.

C Likelihood-only and prior-only reverse anchors

  • On Disagree with LogLin, neither Rlik nor Rprior matches full R on Qwen, Mistral, GLM, or DeepSeek; Llama's Rprior reaches 51.69% versus R at 50.90%.On Llama, Rprior remains weaker than MeanF and GenF as a standalone predictor.Source: C Likelihood-only and prior-only… · p. 12, Table 8 · p. 13
  • Figure 3 plots GTPA@1 by JS rank on the evaluation pool using R, MeanF, and GenF as anchors, with random-agent performance as a dashed line.Source: Figure 3 · p. 13

D Log-linear wR sweep

The appendix tests LogLin’s sensitivity to reverse-anchor weight while retaining wR=0.2 as the main-table default. On the Disagree subset, raising wR can improve accuracy for some backbones.

D Log-linear wR sweep

  • GLM improves from 44.41% at wR=0.2 to 47.57% at wR=0.5 on the Disagree subset.Larger reverse weights can yield higher accuracy on some backbones.Source: D Log-linear wR sweep · p. 13
  • The main tables use wR=0.2 as a fixed light reverse-anchor contribution because FwdJS already incorporates R through JS-based reweighting.The weight is not selected separately for each backbone on the test set.Source: D Log-linear wR sweep · p. 13, Table 9 · p. 13
  • Table 9 reports LogLin GTPA@1 across wR values of 0.2, 0.3, 0.4, and 0.5 on the evaluation pools.The wR=0.2 column matches the LogLin results in the main tables.Source: Table 9 · p. 13, D Log-linear wR sweep · p. 13

E Label collision, co-occurrence, and error-indicator ϕ

On the Disagree subset, π(F, R) remains the lowest of the three label-collision comparisons across all five backbones. The appendix also contrasts correctness co-occurrence with error-indicator association, which measures a different relationship.

E Label collision, co-occurrence, and error-indicator ϕ

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