# Let's Think Var-by-Var: Large Language Models Enable Ad Hoc Probabilistic Reasoning

arXiv: https://arxiv.org/abs/2412.02081 · HTML: https://mlsanity.com/paper/2412.02081/

## Key facts

- Cited by 5 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 5 of its 35 references that are in the map.
- First posted on arXiv 3 Dec 2024.

## 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, 5 of 5)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Why think step by step? Reasoning emerges from the locality of experience](https://mlsanity.com/paper/2304.03843/) (2023)
- [Incoherent Probability Judgments in Large Language Models](https://mlsanity.com/paper/2401.16646/) (2024)
- [BIRD: A Trustworthy Bayesian Inference Framework for Large Language Models](https://mlsanity.com/paper/2404.12494/) (2024)

## Built on by (most cited first, 5 of 5)

- [Modeling Open-World Cognition as On-Demand Synthesis of Probabilistic Models](https://mlsanity.com/paper/2507.12547/) (2025)
- [Always Tell Me The Odds: Fine-grained Conditional Probability Estimation](https://mlsanity.com/paper/2505.01595/) (2025)
- [OpenEstimate: Evaluating LLMs on Reasoning Under Uncertainty with Real-World Data](https://mlsanity.com/paper/2510.15096/) (2025)
- [Probabilistic Reasoning with LLMs for k-anonymity Estimation](https://mlsanity.com/paper/2503.09674/) (2025)
- [Humans and LLMs Diverge on Probabilistic Inferences](https://mlsanity.com/paper/2602.23546/) (2026)

## Abstract (arXiv)

A hallmark of intelligence is the ability to flesh out underspecified situations using "common sense." We propose to extract that common sense from large language models (LLMs), in a form that can feed into probabilistic inference. We focus our investigation on textitguesstimation questions such as "How much are Airbnb listings in Newark, NJ?" Formulating a sensible answer without access to data requires drawing on, and integrating, bits of common knowledge about how textttPrice and textttLocation may relate to other variables, such as textttProperty Type. Our framework answers such a question by synthesizing an textitad hoc probabilistic model. First we prompt an LLM to propose a set of random variables relevant to the question, followed by moment constraints on their joint distribution. We then optimize the joint distribution p within a log-linear family to maximize the overall constraint satisfaction. Our experiments show that LLMs can successfully be prompted to propose reasonable variables, and while the proposed numerical constraints can be noisy, jointly optimizing for their satisfaction reconciles them. When evaluated on probabilistic questions derived from three real-world tabular datasets, we find that our framework performs comparably to a direct prompting baseline in terms of total variation distance from the dataset distribution, and is similarly robust to noise.