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

arXiv: https://arxiv.org/abs/2510.15444 · HTML: https://mlsanity.com/paper/2510.15444/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 2 of its 71 references that are in the map.
- First posted on arXiv 17 Oct 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 2 of 2)

- [Universal Self-Consistency for Large Language Model Generation](https://mlsanity.com/paper/2311.17311/) (2023)
- [Look Before You Leap: An Exploratory Study of Uncertainty Measurement for Large Language Models](https://mlsanity.com/paper/2307.10236/) (2023)

## Built on by (most cited first, 14 of 14)

- [The Six Sigma Agent: Achieving Enterprise-Grade Reliability in LLM Systems Through Consensus-Driven Decomposed Execution](https://mlsanity.com/paper/2601.22290/) (2026)
- [ReThinker: Scientific Reasoning by Rethinking with Guided Reflection and Confidence Control](https://mlsanity.com/paper/2602.04496/) (2026)
- [Illusions of Confidence? Diagnosing LLM Truthfulness via Neighborhood Consistency](https://mlsanity.com/paper/2601.05905/) (2026)
- [NeSy-Route: A Neuro-Symbolic Benchmark for Constrained Route Planning in Remote Sensing](https://mlsanity.com/paper/2603.16307/) (2026)
- [On the Learnability of Test-Time Adaptation: A Recovery Complexity Perspective](https://mlsanity.com/paper/2605.28057/) (2026)
- [Uncovering the Representation Geometry of Minimal Cores in Overcomplete Reasoning Traces](https://mlsanity.com/paper/2605.14358/) (2026)
- [More Correct Mass, Worse Answers: Why Power Sampling Can Fail and How to Fix It](https://mlsanity.com/paper/2608.14420/) (2026)
- [Conditional Performance Guarantee for Large Reasoning Models](https://mlsanity.com/paper/2601.22790/) (2026)
- [Uncertainty Quantification for LLM-based Code Generation](https://mlsanity.com/paper/2605.12201/) (2026)
- [A Self Consistency Based Reranking for Narrative Question Answering](https://mlsanity.com/paper/2606.15741/) (2026)
- [AURORA: Asymmetry and Update-Induced Rotation for Robust Hallucination Detection in Large Language Models](https://mlsanity.com/paper/2606.29545/) (2026)
- [Calibrating Small Language Models for Claim Check-Worthiness Detection](https://mlsanity.com/paper/2608.30731/) (2026)
- [FLEET: From Logits Entropy to Enhanced Trajectories in Text Generation](https://mlsanity.com/paper/2609.27657/) (2026)
- [Compare to Correct: Pairwise-Guided Logit Adjustment for LLM Test-Time Scaling](https://mlsanity.com/paper/2610.18583/) (2026)

## Abstract (arXiv)

Test-time scaling seeks to improve the reasoning performance of large language models (LLMs) by adding computational resources. A prevalent approach within the field is sampling-based test-time scaling methods, which enhance reasoning by generating multiple reasoning paths for a given input during inference. However, despite its practical success, the theoretical foundations remain underexplored. In this paper, we provide the first theoretical framework for analyzing sampling-based test-time scaling methods, grounded in the perspective of confidence estimation. Based on the framework, we analyze two dominant paradigms: self-consistency and perplexity, and reveal key limitations: self-consistency suffers from high estimation error while perplexity exhibits substantial modeling error and possible degradation of the estimation error convergence. To address these limitations, we introduce RPC, a hybrid method that leverages our theoretical insights through two key components: Perplexity Consistency and Reasoning Pruning. Perplexity Consistency combines the strengths of self-consistency and perplexity, boosting the convergence rate of estimation error from linear to exponential while preserving model error. Reasoning Pruning prevents degradation by eliminating low-probability reasoning paths. Both theoretical analysis and empirical results across seven benchmark datasets demonstrate that RPC has a strong potential for reducing reasoning error. Notably, RPC achieves reasoning performance comparable to self-consistency while not only enhancing confidence reliability but also reducing sampling costs by 50%. The code and resources are available at https://wnjxyk.github.io/RPC.