Estimation of probability densities using scale-free field theories

Area of Science:

• Statistics
• Physics
• Data Science

Background:

• Estimating continuous probability density from finite data is a key challenge.
• Statistical field theory offers a promising framework for this problem.
• Existing methods may lack practicality for routine data analysis.

Purpose of the Study:

• To develop a computationally efficient and practical method for probability density estimation.
• To leverage statistical field theory for robust data analysis.
• To create a method that learns smoothness naturally from data.

Main Methods:

• Utilizing statistical field theory for probability density estimation.
• Developing rapid and deterministic computation methods for low dimensions.
• Implementing a data-driven approach to learn smoothness without pre-defined scales.

Main Results:

• A practical, field-theoretic approach for probability density estimation is now available.
• The method allows for rapid and deterministic computation in low dimensions.
• The approach learns a natural smoothness length scale from data via a goodness-of-fit and Occam factor tradeoff.

Conclusions:

• The presented method makes advanced statistical field theory accessible for everyday data analysis.
• This technique offers a powerful alternative for continuous probability density estimation.
• Open-source software is available for one- and two-dimensional applications.