Field Theories for Learning Probability Distributions

TL;DR

This paper explores how field theories can be used as priors over probability distributions, providing a tractable Bayesian framework for inferring distributions from samples, especially in one dimension.

Contribution

It introduces a novel approach using scalar field theories as priors on distributions, with a focus on free field theory in one dimension and potential extensions to higher dimensions.

Findings

Formulation of distribution inference as a scalar field theory problem

Tractable solutions in one-dimensional cases

Discussion of generalizations to higher dimensions

Abstract

Imagine being shown $N$ samples of random variables drawn independently from the same distribution. What can you say about the distribution? In general, of course, the answer is nothing, unless we have some prior notions about what to expect. From a Bayesian point of view we need an {\it a priori} distribution on the space of possible probability distributions, which defines a scalar field theory. In one dimension, free field theory with a constraint provides a tractable formulation of the problem, and we also discus generalizations to higher dimensions.