# Rapid and deterministic estimation of probability densities using scale-free field theories

arXiv: https://arxiv.org/abs/1312.6661 · HTML: https://mlsanity.com/paper/1312.6661/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2013.

## Built on by (most cited first, 2 of 2)

- [Bayesian Cram'er-Rao Bound Estimation with Score-Based Models](https://mlsanity.com/paper/2309.16076/) (2023)
- [Causal, Bayesian, & Non-parametric Modeling of the SARS-CoV-2 Viral Load Distribution vs. Patient's Age](https://mlsanity.com/paper/2105.13483/) (2021)

## Abstract (arXiv)

The question of how best to estimate a continuous probability density from finite data is an intriguing open problem at the interface of statistics and physics. Previous work has argued that this problem can be addressed in a natural way using methods from statistical field theory. Here I describe new results that allow this field-theoretic approach to be rapidly and deterministically computed in low dimensions, making it practical for use in day-to-day data analysis. Importantly, this approach does not impose a privileged length scale for smoothness of the inferred probability density, but rather learns a natural length scale from the data due to the tradeoff between goodness-of-fit and an Occam factor. Open source software implementing this method in one and two dimensions is provided.