Measurement Invariance, Entropy, and Probability

TL;DR

This paper extends maximum entropy methods by incorporating measurement scale as an information constraint, revealing how different scales lead to common probability distributions like Student's and gamma, which explain natural phenomena.

Contribution

It introduces a novel approach that integrates measurement scale into maximum entropy, linking scale transformations to well-known probability distributions and unifying concepts like superstatistics.

Findings

Measurement scale determines the resulting probability distribution.

Linear-log and inverse scales lead to Student's and gamma distributions.

Superstatistics is a special case of measurement scale transformations.

Abstract

We show that the natural scaling of measurement for a particular problem defines the most likely probability distribution of observations taken from that measurement scale. Our approach extends the method of maximum entropy to use measurement scale as a type of information constraint. We argue that a very common measurement scale is linear at small magnitudes grading into logarithmic at large magnitudes, leading to observations that often follow Student's probability distribution which has a Gaussian shape for small fluctuations from the mean and a power law shape for large fluctuations from the mean. An inverse scaling often arises in which measures naturally grade from logarithmic to linear as one moves from small to large magnitudes, leading to observations that often follow a gamma probability distribution. A gamma distribution has a power law shape for small magnitudes and an exponential shape for large magnitudes. The two measurement scales are natural inverses connected by the Laplace integral transform. This inversion connects the two major scaling patterns commonly found in nature. We also show that superstatistics is a special case of an integral transform, and thus can be understood as a particular way in which to change the scale of measurement. Incorporating information about measurement scale into maximum entropy provides a general approach to the relations between measurement, information and probability.