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From Drift to Coherence: Stabilizing Beliefs in LLMs

ICML 2026 · Large language models › Reasoning and chain of thought · approach: Test-time compute scaling · 0 citations (OpenAlex)

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Shows LLM answer resampling on MCQA drifts early but self-stabilizes, then proposes answer seeding and a self-consistency loss (Martingale Amortization) to reduce drift and improve calibration.

Problem

Whether LLMs satisfy the martingale (coherence) property of predictive beliefs in realistic generic question answering, where prior work only tested synthetic ICL tasks; LLM answer sequences show early belief drift.

Approach

Prompted predictive resampling asks the LLM to generate a sequence of answers to the same question; the belief drift is measured exactly over the discrete answer simplex. Answer seeding (prepending i.i.d. seed answers) speeds stabilization, and a self-consistency loss fine-tunes the model so its short-horizon predictions match its long-horizon ones, amortizing burn-in into zero-shot predictions.

Key results

• On Llama-3.1-8B, AM-PPR + SEED_5 reduces CSQA ECE to 0.0674 from 0.1072 for Direct-Query and gives 77.0% accuracy vs 78.0% for Direct-Query.
• AM-PPR + SEED_1 reaches 62.3% on TinyMMLU and 81.5% on AI2-ARC accuracy, comparable to Direct-Query (61.2% and 80.5%).
• Answer seeding is needed for PPR accuracy: PPR + NoSEED gets 50.0% on CSQA versus 74.0% with one seed.

Surprising findings

• Lower martingale drift does not monotonically improve calibration; the authors note the martingale property is only a necessary condition for good calibration.
• AM-PPR + SEED_1 has much higher NLL (2.2144) than PPR on TinyMMLU despite lower ECE.

Limitations

• Martingale property is only necessary, not sufficient, for calibrated Bayesian predictions.
• Requires prompt engineering to elicit clean sequential resampling output.
• Theoretical guarantees rely on assumptions (uniform consistency, KL bound) that hold only over a finite horizon set; implementation uses a subset of horizon pairs and an ad hoc policy-gradient correction.

Experimental setup

Datasets: CSQA

Benchmarks: CSQA AI2-ARC TinyMMLU

Metrics: Accuracy AUC ECE NLL Brier score expected martingale drift (EMD)

Baselines: PPR + NoSEED PPR + SEED_1 PPR + SEED_5 Direct-Query

Builds on: martingale posterior conditionally identically distributed sequences LLM martingale violation analysis (prior work)

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