# Verbalized Particle Posterior: Bayesian Inference over Natural Language Hypotheses

arXiv: https://arxiv.org/abs/2607.22961 · HTML: https://mlsanity.com/paper/2607.22961/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 41 references that are in the map.
- First posted on arXiv 25 Jul 2026.

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

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [Voyager: An Open-Ended Embodied Agent with Large Language Models](https://mlsanity.com/paper/2305.16291/) (2023)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Large Language Models Are Human-Level Prompt Engineers](https://mlsanity.com/paper/2211.01910/) (2022)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Large Language Models as Optimizers](https://mlsanity.com/paper/2309.03409/) (2023)
- [Automatic Prompt Optimization with "Gradient Descent" and Beam Search](https://mlsanity.com/paper/2305.03495/) (2023)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [TextGrad: Automatic "Differentiation" via Text](https://mlsanity.com/paper/2406.07496/) (2024)
- [Conformal Language Modeling](https://mlsanity.com/paper/2306.10193/) (2023)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Hypothesis Search: Inductive Reasoning with Language Models](https://mlsanity.com/paper/2309.05660/) (2023)
- [CGMH: Constrained Sentence Generation by Metropolis-Hastings Sampling](https://mlsanity.com/paper/1811.10996/) (2018)
- [Phenomenal Yet Puzzling: Testing Inductive Reasoning Capabilities of Language Models with Hypothesis Refinement](https://mlsanity.com/paper/2310.08559/) (2023)
- [Symbolic Regression with a Learned Concept Library](https://mlsanity.com/paper/2409.09359/) (2024)
- [LLM Processes: Numerical Predictive Distributions Conditioned on Natural Language](https://mlsanity.com/paper/2405.12856/) (2024)
- [Quality-Aware Decoding for Neural Machine Translation](https://mlsanity.com/paper/2205.00978/) (2022)
- [In-Context Symbolic Regression: Leveraging Large Language Models for Function Discovery](https://mlsanity.com/paper/2404.19094/) (2024)
- [Verbalized Machine Learning: Revisiting Machine Learning with Language Models](https://mlsanity.com/paper/2406.04344/) (2024)
- [Bayesian Concept Bottleneck Models with LLM Priors](https://mlsanity.com/paper/2410.15555/) (2024)
- [BayesAgent: Bayesian Agentic Reasoning Under Uncertainty via Verbalized Probabilistic Graphical Modeling](https://mlsanity.com/paper/2406.05516/) (2024)

## Abstract (arXiv)

Verbalized Machine Learning (VML) parameterizes a model as a natural-language prompt that an LLM evaluates as f(x; theta). The framework is interpretable, but it commits to a single hypothesis with no measure of uncertainty, and that hypothesis varies substantially across optimization runs on the same data. We propose the Verbalized Particle Posterior (VPP), which treats verbalized learning as a Bayesian inference problem: maintain a population of natural-language hypotheses as particles, update them with Metropolis-Hastings (VPP-MH) or Sequential Monte Carlo (VPP-SMC), and predict by Bayesian model averaging. Both algorithms treat the LLM as a black box, requiring no access to logits or gradients. A distinctive consequence follows. In classical Bayesian learning, model selection sits outside the posterior; in VPP both model structure and parameters share a single language space, and the posterior ranges over both. We evaluate VPP on regression, classification, and rule-discovery benchmarks. It improves over a single VML run on every benchmark and matches or exceeds an oracle-best ensemble of independent VML runs on most, while eliminating the catastrophic single-run failures that VML occasionally produces. Because each particle is a human-readable hypothesis, the posterior is itself something a reader can inspect, seeing in plain text which explanations the data supported and which it ruled out.