# Density estimation with LLMs: a geometric investigation of in-context learning trajectories

arXiv: https://arxiv.org/abs/2410.05218 · HTML: https://mlsanity.com/paper/2410.05218/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 39 references that are in the map.
- First posted on arXiv 7 Oct 2024.

## Builds on (most cited first, 15 of 15)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [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)
- [Why Can GPT Learn In-Context? Language Models Implicitly Perform Gradient Descent as Meta-Optimizers](https://mlsanity.com/paper/2212.10559/) (2022)
- [Birth of a Transformer: A Memory Viewpoint](https://mlsanity.com/paper/2306.00802/) (2023)
- [LLM Processes: Numerical Predictive Distributions Conditioned on Natural Language](https://mlsanity.com/paper/2405.12856/) (2024)
- [Measuring Inductive Biases of In-Context Learning with Underspecified Demonstrations](https://mlsanity.com/paper/2305.13299/) (2023)
- [LLMs learn governing principles of dynamical systems, revealing an in-context neural scaling law](https://mlsanity.com/paper/2402.00795/) (2024)
- [The Training Process of Many Deep Networks Explores the Same Low-Dimensional Manifold](https://mlsanity.com/paper/2305.01604/) (2023)
- [How Do Transformers "Do" Physics? Investigating the Simple Harmonic Oscillator](https://mlsanity.com/paper/2405.17209/) (2024)
- [Can LLMs predict the convergence of Stochastic Gradient Descent?](https://mlsanity.com/paper/2408.01736/) (2024)

## Built on by (most cited first, 6 of 6)

- [The Shape of Beliefs: Geometry, Dynamics, and Interventions along Representation Manifolds of Language Models' Posteriors](https://mlsanity.com/paper/2602.02315/) (2026)
- [Text-Trained LLMs Can Zero-Shot Extrapolate PDE Dynamics, Revealing a Three-Stage In-Context Learning Mechanism](https://mlsanity.com/paper/2509.06322/) (2025)
- [Pre-trained Large Language Models Learn Hidden Markov Models In-context](https://mlsanity.com/paper/2506.07298/) (2025)
- [Jacobian Scopes: token-level causal attributions in LLMs](https://mlsanity.com/paper/2601.16407/) (2026)
- [Transformers as Unsupervised Learning Algorithms: A study on Gaussian Mixtures](https://mlsanity.com/paper/2505.11918/) (2025)
- [A Graph Signal Processing Perspective on Numerical Sequence Representations in LLM In-Context Learning](https://mlsanity.com/paper/2608.03015/) (2026)

## Abstract (arXiv)

Large language models (LLMs) demonstrate remarkable emergent abilities to perform in-context learning across various tasks, including time series forecasting. This work investigates LLMs' ability to estimate probability density functions (PDFs) from data observed in-context; such density estimation (DE) is a fundamental task underlying many probabilistic modeling problems. We leverage the Intensive Principal Component Analysis (InPCA) to visualize and analyze the in-context learning dynamics of LLaMA-2 models. Our main finding is that these LLMs all follow similar learning trajectories in a low-dimensional InPCA space, which are distinct from those of traditional density estimation methods like histograms and Gaussian kernel density estimation (KDE). We interpret the LLaMA in-context DE process as a KDE with an adaptive kernel width and shape. This custom kernel model captures a significant portion of LLaMA's behavior despite having only two parameters. We further speculate on why LLaMA's kernel width and shape differs from classical algorithms, providing insights into the mechanism of in-context probabilistic reasoning in LLMs. Our codebase, along with a 3D visualization of an LLM's in-context learning trajectory, is publicly available at https://github.com/AntonioLiu97/LLMICL_inPCA