# Enough Coin Flips Can Make LLMs Act Bayesian

arXiv: https://arxiv.org/abs/2503.04722 · HTML: https://mlsanity.com/paper/2503.04722/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 38 references that are in the map.
- First posted on arXiv 6 Mar 2025.

## Builds on (most cited first, 28 of 28)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Generative Agents: Interactive Simulacra of Human Behavior](https://mlsanity.com/paper/2304.03442/) (2023)
- [Lost in the Middle: How Language Models Use Long Contexts](https://mlsanity.com/paper/2307.03172/) (2023)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone](https://mlsanity.com/paper/2404.14219/) (2024)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (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)
- [Just Ask for Calibration: Strategies for Eliciting Calibrated Confidence Scores from Language Models Fine-Tuned with Human Feedback](https://mlsanity.com/paper/2305.14975/) (2023)
- [Out of One, Many: Using Language Models to Simulate Human Samples](https://mlsanity.com/paper/2209.06899/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Using Large Language Models to Simulate Multiple Humans and Replicate Human Subject Studies](https://mlsanity.com/paper/2208.10264/) (2022)
- [OLMoE: Open Mixture-of-Experts Language Models](https://mlsanity.com/paper/2409.02060/) (2024)
- [AI Chains: Transparent and Controllable Human-AI Interaction by Chaining Large Language Model Prompts](https://mlsanity.com/paper/2110.01691/) (2021)
- [Trial and Error: Exploration-Based Trajectory Optimization for LLM Agents](https://mlsanity.com/paper/2403.02502/) (2024)
- [DriveDreamer-2: LLM-Enhanced World Models for Diverse Driving Video Generation](https://mlsanity.com/paper/2403.06845/) (2024)
- [Large Language Models Are Latent Variable Models: Explaining and Finding Good Demonstrations for In-Context Learning](https://mlsanity.com/paper/2301.11916/) (2023)
- [What and How does In-Context Learning Learn? Bayesian Model Averaging, Parameterization, and Generalization](https://mlsanity.com/paper/2305.19420/) (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)
- [Is In-Context Learning in Large Language Models Bayesian? A Martingale Perspective](https://mlsanity.com/paper/2406.00793/) (2024)
- [Virtual Personas for Language Models via an Anthology of Backstories](https://mlsanity.com/paper/2407.06576/) (2024)
- [ALOHa: A New Measure for Hallucination in Captioning Models](https://mlsanity.com/paper/2404.02904/) (2024)
- [LLMs learn governing principles of dynamical systems, revealing an in-context neural scaling law](https://mlsanity.com/paper/2402.00795/) (2024)
- [Heads, Tails, and AI Fails: LLMs, Randomness, and Human Judgments](https://mlsanity.com/paper/2406.00092/) (2024)
- [Do LLMs Play Dice? Exploring Probability Distribution Sampling in Large Language Models for Behavioral Simulation](https://mlsanity.com/paper/2404.09043/) (2024)

## Built on by (most cited first, 22 of 22)

- [String Seed of Thought: Prompting LLMs for Distribution-Faithful and Diverse Generation](https://mlsanity.com/paper/2510.21150/) (2025)
- [The Illusion of Stochasticity in LLMs](https://mlsanity.com/paper/2604.06543/) (2026)
- [Flipping Against All Odds: Reducing LLM Coin Flip Bias via Verbalized Rejection Sampling](https://mlsanity.com/paper/2506.09998/) (2025)
- [Martingale Score: An Unsupervised Metric for Bayesian Rationality in LLM Reasoning](https://mlsanity.com/paper/2512.02914/) (2025)
- [Pre-trained Large Language Models Learn Hidden Markov Models In-context](https://mlsanity.com/paper/2506.07298/) (2025)
- [Position: agentic AI orchestration should be Bayes-consistent](https://mlsanity.com/paper/2605.00742/) (2026)
- [Beyond Scaling Curves: Internal Dynamics of Neural Networks Through the NTK Lens](https://mlsanity.com/paper/2507.05035/) (2025)
- [AdaptFuse: Training-Free Sequential Preference Inference via Externalized Bayesian Computation](https://mlsanity.com/paper/2604.03925/) (2026)
- [Express Your Doubts -- Probabilistic World Modeling Should not be Based on Token logprobs](https://mlsanity.com/paper/2505.02072/) (2025)
- [SPIKE-RL: Video-LLMs meet Bayesian Surprise](https://mlsanity.com/paper/2509.23433/) (2025)
- [TextBandit: Evaluating Probabilistic Reasoning in LLMs Through Language-Only Decision Tasks](https://mlsanity.com/paper/2510.13878/) (2025)
- [D-Models and E-Models: Diversity-Stability Trade-offs in the Sampling Behavior of Large Language Models](https://mlsanity.com/paper/2601.17865/) (2026)
- [Large Language Model for Discrete Optimization Problems: Evaluation and Step-by-step Reasoning](https://mlsanity.com/paper/2603.07733/) (2026)
- [Beyond Facts: Benchmarking Distributional Reading Comprehension in Large Language Models](https://mlsanity.com/paper/2604.06201/) (2026)
- [EviCare: Enhancing Diagnosis Prediction with Deep Model-Guided Evidence for In-Context Reasoning](https://mlsanity.com/paper/2604.10455/) (2026)
- [Mutual Debiasing via Dual-Seed Comparison for Probabilistic Sampling in Large Language Models](https://mlsanity.com/paper/2608.26161/) (2026)
- [From Static Personal Values to Contextualized Personalization: Bayesian Personalized Value Alignment for LLMs](https://mlsanity.com/paper/2609.28942/) (2026)
- [Small transformers track Bayesian evidence for latent common causes via a context-invariant mechanism](https://mlsanity.com/paper/2609.35161/) (2026)
- [Diagnosing and Improving Probabilistic Reasoning in Large Language Models](https://mlsanity.com/paper/2609.38005/) (2026)
- [Bayesian Fine-tuning Yields Language Models that are as Bayesian as their Beliefs Allow](https://mlsanity.com/paper/2610.00679/) (2026)
- [Do LLMs Understand Sequential Structure? A Controlled Study of Inference and Generation](https://mlsanity.com/paper/2610.04977/) (2026)
- [Copies or Sources? Measuring How LLM Aggregators Count Restated Evidence in Multi-Agent Systems](https://mlsanity.com/paper/2610.06192/) (2026)

## Abstract (arXiv)

Large language models (LLMs) exhibit the ability to generalize given few-shot examples in their input prompt, an emergent capability known as in-context learning (ICL). We investigate whether LLMs use ICL to perform structured reasoning in ways that are consistent with a Bayesian framework or rely on pattern matching. Using a controlled setting of biased coin flips, we find that: (1) LLMs often possess biased priors, causing initial divergence in zero-shot settings, (2) in-context evidence outweighs explicit bias instructions, (3) LLMs broadly follow Bayesian posterior updates, with deviations primarily due to miscalibrated priors rather than flawed updates, and (4) attention magnitude has negligible effect on Bayesian inference. With sufficient demonstrations of biased coin flips via ICL, LLMs update their priors in a Bayesian manner.