# Probability Consistency in Large Language Models: Theoretical Foundations Meet Empirical Discrepancies

arXiv: https://arxiv.org/abs/2505.08739 · HTML: https://mlsanity.com/paper/2505.08739/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 37 references that are in the map.
- First posted on arXiv 13 May 2025.

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Lost in the Middle: How Language Models Use Long Contexts](https://mlsanity.com/paper/2307.03172/) (2023)
- [Neural Machine Translation of Rare Words with Subword Units](https://mlsanity.com/paper/1508.07909/) (2015)
- [XLNet: Generalized Autoregressive Pretraining for Language Understanding](https://mlsanity.com/paper/1906.08237/) (2019)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [What Does BERT Look At? An Analysis of BERT's Attention](https://mlsanity.com/paper/1906.04341/) (2019)
- [Measuring Faithfulness in Chain-of-Thought Reasoning](https://mlsanity.com/paper/2307.13702/) (2023)
- [Smarter, Better, Faster, Longer: A Modern Bidirectional Encoder for Fast, Memory Efficient, and Long Context Finetuning and Inference](https://mlsanity.com/paper/2412.13663/) (2024)
- [Reasoning Models Don't Always Say What They Think](https://mlsanity.com/paper/2505.05410/) (2025)
- [Revealing the Dark Secrets of BERT](https://mlsanity.com/paper/1908.08593/) (2019)
- [cosFormer: Rethinking Softmax in Attention](https://mlsanity.com/paper/2202.08791/) (2022)
- [When Attention Sink Emerges in Language Models: An Empirical View](https://mlsanity.com/paper/2410.10781/) (2024)
- [Quantizable Transformers: Removing Outliers by Helping Attention Heads Do Nothing](https://mlsanity.com/paper/2306.12929/) (2023)
- [Transformers need glasses! Information over-squashing in language tasks](https://mlsanity.com/paper/2406.04267/) (2024)
- [Unveiling and Harnessing Hidden Attention Sinks: Enhancing Large Language Models without Training through Attention Calibration](https://mlsanity.com/paper/2406.15765/) (2024)
- [Large language models surpass human experts in predicting neuroscience results](https://mlsanity.com/paper/2403.03230/) (2024)
- [On the Emergence of Position Bias in Transformers](https://mlsanity.com/paper/2502.01951/) (2025)
- [Regularizing Neural Machine Translation by Target-bidirectional Agreement](https://mlsanity.com/paper/1808.04064/) (2018)
- [How do language models learn facts? Dynamics, curricula and hallucinations](https://mlsanity.com/paper/2503.21676/) (2025)
- [Mission: Impossible Language Models](https://mlsanity.com/paper/2401.06416/) (2024)
- [Learned feature representations are biased by complexity, learning order, position, and more](https://mlsanity.com/paper/2405.05847/) (2024)
- [Meet in the Middle: A New Pre-training Paradigm](https://mlsanity.com/paper/2303.07295/) (2023)
- [Arrows of Time for Large Language Models](https://mlsanity.com/paper/2401.17505/) (2024)
- [Matching domain experts by training from scratch on domain knowledge](https://mlsanity.com/paper/2405.09395/) (2024)
- [Reverse Modeling in Large Language Models](https://mlsanity.com/paper/2410.09817/) (2024)
- [Confidence-weighted integration of human and machine judgments for superior decision-making](https://mlsanity.com/paper/2408.08083/) (2024)

## Built on by (most cited first, 1 of 1)

- [Do Tabular Foundation Models Agree with Themselves?](https://mlsanity.com/paper/2608.06004/) (2026)

## Abstract (arXiv)

Can autoregressive large language models (LLMs) learn consistent probability distributions when trained on sequences in different token orders? We prove formally that for any well-defined probability distribution, sequence perplexity is invariant under any factorization, including forward, backward, or arbitrary permutations. This result establishes a rigorous theoretical foundation for studying how LLMs learn from data and defines principled protocols for empirical evaluation. Applying these protocols, we show that prior studies examining ordering effects suffer from critical methodological flaws. We retrain GPT-2 models across forward, backward, and arbitrary permuted orders on scientific text. We find systematic deviations from theoretical invariance across all orderings with arbitrary permutations strongly deviating from both forward and backward models, which largely (but not completely) agreed with one another. Deviations were traceable to differences in self-attention, reflecting positional and locality biases in processing. Our theoretical and empirical results provide novel avenues for understanding positional biases in LLMs and suggest methods for detecting when LLMs' probability distributions are inconsistent and therefore untrustworthy.