# Is In-Context Learning in Large Language Models Bayesian? A Martingale Perspective

arXiv: https://arxiv.org/abs/2406.00793 · HTML: https://mlsanity.com/paper/2406.00793/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 55 references that are in the map.
- First posted on arXiv 2 Jun 2024.

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

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [Calibrate Before Use: Improving Few-Shot Performance of Language Models](https://mlsanity.com/paper/2102.09690/) (2021)
- [Fantastically Ordered Prompts and Where to Find Them: Overcoming Few-Shot Prompt Order Sensitivity](https://mlsanity.com/paper/2104.08786/) (2021)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [Time-LLM: Time Series Forecasting by Reprogramming Large Language Models](https://mlsanity.com/paper/2310.01728/) (2023)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Large Language Models Are Zero-Shot Time Series Forecasters](https://mlsanity.com/paper/2310.07820/) (2023)
- [MetaICL: Learning to Learn In Context](https://mlsanity.com/paper/2110.15943/) (2021)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)
- [Language Models are Realistic Tabular Data Generators](https://mlsanity.com/paper/2210.06280/) (2022)
- [Does Synthetic Data Generation of LLMs Help Clinical Text Mining?](https://mlsanity.com/paper/2303.04360/) (2023)
- [Large Language Models Are Latent Variable Models: Explaining and Finding Good Demonstrations for In-Context Learning](https://mlsanity.com/paper/2301.11916/) (2023)
- [Pretraining task diversity and the emergence of non-Bayesian in-context learning for regression](https://mlsanity.com/paper/2306.15063/) (2023)
- [What and How does In-Context Learning Learn? Bayesian Model Averaging, Parameterization, and Generalization](https://mlsanity.com/paper/2305.19420/) (2023)
- [Calibrated Language Models Must Hallucinate](https://mlsanity.com/paper/2311.14648/) (2023)
- [Uncertainty Quantification with Pre-trained Language Models: A Large-Scale Empirical Analysis](https://mlsanity.com/paper/2210.04714/) (2022)
- [Synthetic Data Generation with Large Language Models for Text Classification: Potential and Limitations](https://mlsanity.com/paper/2310.07849/) (2023)
- [A Theory of Emergent In-Context Learning as Implicit Structure Induction](https://mlsanity.com/paper/2303.07971/) (2023)
- [In-Context Learning through the Bayesian Prism](https://mlsanity.com/paper/2306.04891/) (2023)
- [The Transient Nature of Emergent In-Context Learning in Transformers](https://mlsanity.com/paper/2311.08360/) (2023)
- [Making Pre-trained Language Models Great on Tabular Prediction](https://mlsanity.com/paper/2403.01841/) (2024)
- [Martingale posterior distributions](https://mlsanity.com/paper/2103.15671/) (2021)
- [A Latent Space Theory for Emergent Abilities in Large Language Models](https://mlsanity.com/paper/2304.09960/) (2023)
- [Generating Faithful Synthetic Data with Large Language Models: A Case Study in Computational Social Science](https://mlsanity.com/paper/2305.15041/) (2023)

## Built on by (most cited first, 30 of 57)

- [Plan and Budget: Effective and Efficient Test-Time Scaling on Reasoning Large Language Models](https://mlsanity.com/paper/2505.16122/) (2025)
- [BED-LLM: Intelligent Information Gathering with LLMs and Bayesian Experimental Design](https://mlsanity.com/paper/2508.21184/) (2025)
- [Enough Coin Flips Can Make LLMs Act Bayesian](https://mlsanity.com/paper/2503.04722/) (2025)
- [Estimating the Hallucination Rate of Generative AI](https://mlsanity.com/paper/2406.07457/) (2024)
- [Position: The Future of Bayesian Prediction Is Prior-Fitted](https://mlsanity.com/paper/2505.23947/) (2025)
- [Bayesian scaling laws for in-context learning](https://mlsanity.com/paper/2410.16531/) (2024)
- [Bayesian Concept Bottleneck Models with LLM Priors](https://mlsanity.com/paper/2410.15555/) (2024)
- [AutoElicit: Using Large Language Models for Expert Prior Elicitation in Predictive Modelling](https://mlsanity.com/paper/2411.17284/) (2024)
- [Everything Everywhere All at Once: LLMs can In-Context Learn Multiple Tasks in Superposition](https://mlsanity.com/paper/2410.05603/) (2024)
- [Are Large Language Models Reliable AI Scientists? Assessing Reverse-Engineering of Black-Box Systems](https://mlsanity.com/paper/2505.17968/) (2025)
- [Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations](https://mlsanity.com/paper/2509.05186/) (2025)
- [Agentic Forecasting using Sequential Bayesian Updating of Linguistic Beliefs](https://mlsanity.com/paper/2604.18576/) (2026)
- [TabMGP: Martingale Posterior with TabPFN](https://mlsanity.com/paper/2510.25154/) (2025)
- [From Zero to Hero: Advancing Zero-Shot Foundation Models for Tabular Outlier Detection](https://mlsanity.com/paper/2602.03018/) (2026)
- [Transformers Handle Endogeneity in In-Context Linear Regression](https://mlsanity.com/paper/2410.01265/) (2024)
- [Uncertainty Quantification for Prior-Data Fitted Networks using Martingale Posteriors](https://mlsanity.com/paper/2505.11325/) (2025)
- [Martingale Score: An Unsupervised Metric for Bayesian Rationality in LLM Reasoning](https://mlsanity.com/paper/2512.02914/) (2025)
- [Verbalizing LLM's Higher-order Uncertainty via Imprecise Probabilities](https://mlsanity.com/paper/2603.10396/) (2026)
- [DeFine: Decision-Making with Analogical Reasoning over Factor Profiles](https://mlsanity.com/paper/2410.01772/) (2024)
- [In-Context Parametric Inference: Point or Distribution Estimators?](https://mlsanity.com/paper/2502.11617/) (2025)
- [Counterfactual reasoning: an analysis of in-context emergence](https://mlsanity.com/paper/2506.05188/) (2025)
- [Provable In-Context Learning of Nonlinear Regression with Transformers](https://mlsanity.com/paper/2507.20443/) (2025)
- [The Initialization Determines Whether In-Context Learning Is Gradient Descent](https://mlsanity.com/paper/2512.04268/) (2025)
- [Rethinking Zero-Shot Time Series Classification: From Task-specific Classifiers to In-Context Inference](https://mlsanity.com/paper/2602.00620/) (2026)
- [Position: agentic AI orchestration should be Bayes-consistent](https://mlsanity.com/paper/2605.00742/) (2026)
- [VIP-COP: Context Optimization for Tabular Foundation Models](https://mlsanity.com/paper/2605.12904/) (2026)
- [From Drift to Coherence: Stabilizing Beliefs in LLMs](https://mlsanity.com/paper/2606.17832/) (2026)
- [Can Generative AI Solve Your In-Context Learning Problem? A Martingale Perspective](https://mlsanity.com/paper/2412.06033/) (2024)
- [Architectural and Inferential Inductive Biases For Exchangeable Sequence Modeling](https://mlsanity.com/paper/2503.01215/) (2025)
- [From predictions to confidence intervals: an empirical study of conformal prediction methods for in-context learning](https://mlsanity.com/paper/2504.15722/) (2025)

## Abstract (arXiv)

In-context learning (ICL) has emerged as a particularly remarkable characteristic of Large Language Models (LLM): given a pretrained LLM and an observed dataset, LLMs can make predictions for new data points from the same distribution without fine-tuning. Numerous works have postulated ICL as approximately Bayesian inference, rendering this a natural hypothesis. In this work, we analyse this hypothesis from a new angle through the martingale property, a fundamental requirement of a Bayesian learning system for exchangeable data. We show that the martingale property is a necessary condition for unambiguous predictions in such scenarios, and enables a principled, decomposed notion of uncertainty vital in trustworthy, safety-critical systems. We derive actionable checks with corresponding theory and test statistics which must hold if the martingale property is satisfied. We also examine if uncertainty in LLMs decreases as expected in Bayesian learning when more data is observed. In three experiments, we provide evidence for violations of the martingale property, and deviations from a Bayesian scaling behaviour of uncertainty, falsifying the hypothesis that ICL is Bayesian.