---
title: "Automated Statistical Model Discovery with Language Models"
description: "The authors propose BoxLM, which lets large language models fill both roles within Box's Loop. 44 references, cited by 12 papers here."
authors: ["Michael Y. Li", "Emily B. Fox", "Noah D. Goodman"]
year: 2024
venue: "arXiv:2402.17879"
source: "https://arxiv.org/abs/2402.17879"
doi: "10.48550/arxiv.2402.17879"
arxiv: "2402.17879"
licence: "arXiv non-exclusive distribution license"
licence_url: "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/li-2024-automated-statistical-model-discovery/"
---
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# Automated Statistical Model Discovery with Language Models

**Authors:** [Michael Y. Li](https://md.research.elamur.ai/free/en/authors/michael-y-li/), Emily B. Fox and [Noah D. Goodman](https://md.research.elamur.ai/free/en/authors/noah-d-goodman/) · **Year:** 2024 · **Venue:** arXiv:2402.17879 · **Licence:** [arXiv non-exclusive distribution license](http://arxiv.org/licenses/nonexclusive-distrib/1.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/2402.17879) · [PDF](https://arxiv.org/pdf/2402.17879) · [DOI 10.48550/arxiv.2402.17879](https://doi.org/10.48550/arxiv.2402.17879) · [arXiv 2402.17879](https://arxiv.org/abs/2402.17879) · [OpenAlex](https://openalex.org/W4400021634)

**Citations:** 2 citations (OpenAlex, 2026-10-14) · [12](https://md.research.elamur.ai/free/en/papers/li-2024-automated-statistical-model-discovery/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
Building a good statistical model usually requires both a skilled modeler and a domain expert. Earlier automation systems, such as the Automatic Statistician, only worked inside a hand-designed domain-specific language (DSL) with a fixed search routine. The authors propose BoxLM, which lets large language models fill both roles within Box's Loop. A proposal LM writes candidate models as probabilistic programs. It conditions on the data, its metadata, the best earlier programs and textual feedback. Each program is fitted with generic inference and scored, for instance by ELPD LOO. A critic LM then reads posterior predictive statistics and turns them into plain-language advice for the next iteration. The evaluation covers three scenarios: search within a DSL of Gaussian process kernels, open-ended modeling of real datasets, and refining classic expert models under constraints stated in natural language. BoxLM performs on par with the Automatic Statistician and with programs written by human experts, and it extends Lotka-Volterra dynamics in interpretable ways. Because no DSL or custom search has to be engineered, model discovery could become more accessible.

## Key points
- Published at ICML 2024 (PMLR 235) by Stanford researchers. The system, BoxLM, generates m programs per round, keeps the top-k as exemplars, and uses critic-LM feedback to steer the next round.
- In an illustrative dataset of 27 dugongs (age vs. length), the critic observes that programs using the von Bertalanffy growth function obtain better LOO scores. It also recommends informative, biology-based priors.
- In the Stan experiments, scores averaged over 3 rounds of Box's Loop improve from round to round on average, though not monotonically for every individual run.
- On Lotka-Volterra predator-prey data, all BoxLM variants (No-WS, WS-Constraint, WS-No Constraint) beat a standard Lotka-Volterra baseline and a hybrid neural ODE baseline on test MAE, averaged over three runs. Under an interpretability constraint, BoxLM extended the Rosenzweig–MacArthur model with Holling type II response by making handling time an MLP function of prey density.
- Stated limitations:
  - only static, one-dimensional datasets were studied;
  - criticism statistics (residuals, posterior predictive mean) were fixed in advance;
  - the method relies on in-context learning rather than fine-tuning.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

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## Cite

Cite as: Michael Y. Li, Emily B. Fox and Noah D. Goodman. 2024. Automated Statistical Model Discovery with Language Models. arXiv:2402.17879.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{li2024automated,
  title = {{Automated Statistical Model Discovery with Language Models}},
  author = {Michael Y. Li, Emily B. Fox and Noah D. Goodman},
  year = {2024},
  doi = {10.48550/arxiv.2402.17879},
  eprint = {2402.17879},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2402.17879}
}
```

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*Updated 3 November 2026*