# Extraction of physical laws from joint experimental data

## TL;DR

This paper presents a method for extracting physical laws from experimental data using probabilistic estimators, information theory, and a cost function to determine optimal data quantity for accurate law identification.

## Contribution

It introduces a new estimator based on conditional averages, a novel prediction quality measure, and a cost function combining redundancy and discrepancy for optimal data selection.

## Key findings

- The proposed estimator effectively captures the physical law from data.
- Redundancy increases with more experiments, while information converges to a limit.
- The method distinguishes independent variables using mutual information ratios.

## Abstract

The extraction of a physical law y=yo(x) from joint experimental data about x and y is treated. The joint, the marginal and the conditional probability density functions (PDF) are expressed by given data over an estimator whose kernel is the instrument scattering function. As an optimal estimator of yo(x) the conditional average is proposed. The analysis of its properties is based upon a new definition of prediction quality. The joint experimental information and the redundancy of joint measurements are expressed by the relative entropy. With the number of experiments the redundancy on average increases, while the experimental information converges to a certain limit value. The difference between this limit value and the experimental information at a finite number of data represents the discrepancy between the experimentally determined and the true properties of the phenomenon. The sum of the discrepancy measure and the redundancy is utilized as a cost function. By its minimum a reasonable number of data for the extraction of the law yo(x) is specified. The mutual information is defined by the marginal and the conditional PDFs of the variables. The ratio between mutual information and marginal information is used to indicate which variable is the independent one. The properties of the introduced statistics are demonstrated on deterministically and randomly related variables.