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A Theoretical Study on Bridging Internal Probability and Self-Consistency for LLM Reasoning

NeurIPS 2025 · Large language models › Reasoning and chain of thought · approach: Test-time compute scaling · 1 citations (OpenAlex)

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Proposes a theoretical error decomposition (estimation vs. model error) for sampling-based test-time scaling and the RPC method, which combines perplexity with self-consistency (Perplexity Consistency) plus probability-based Reasoning Pruning via Weibull mixture modeling.

Problem

Sampling-based test-time scaling (self-consistency, perplexity) for LLM reasoning lacks theoretical grounding; self-consistency converges slowly in estimation error and perplexity has large model error and degrading convergence when path probabilities are low.

Approach

Decompose reasoning error into estimation and model error under a confidence-estimation framework, then fuse internal path probabilities into self-consistency (Perplexity Consistency) for exponential estimation-error convergence and prune low-probability paths with a Weibull-mixture threshold (Reasoning Pruning).

Key results

• RPC on InternLM2-Math-Plus 7B averages 26.11% accuracy vs 24.82% for SC across four math benchmarks, with average ECE 12.37 vs 13.37.
• RPC matches best SC accuracy on MATH with 32 samples instead of 64 (50% reduction) and on AIME with 48 instead of 128 samples (62.5% reduction).
• On MathOdyssey with 7B, RPC reaches 31.62% vs 28.25% for SC, the largest reported gain.

Measured effects reported in the paper

Surprising findings

• Perplexity Consistency alone shows only marginal gains over SC on several benchmarks, and the Reasoning Pruning step accounts for much of the improvement, with perplexity alone performing far below SC on several math tasks.

Limitations

• Theoretical analysis relies on idealized assumptions such as Bernoulli sampling and distinct incorrect answers.
• Pruning threshold estimated per problem from a Weibull mixture may be unstable when the number of samples is small.
• Experiments use only a few open models and sample sizes up to 128 per problem.

Experimental setup

Datasets: MATH MathOdyssey OlympiadBench AIME HumanEval MBPP APPS GPQA LogiQA

Benchmarks: MATH MathOdyssey OlympiadBench AIME HumanEval MBPP APPS GPQA LogiQA

Metrics: accuracy expected calibration error sampling budget to match SC

Baselines: PPL SC Verbalized confidence

Builds on: self-consistency perplexity Enhanced self-consistency (ESC) Truncated mean

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