Lessons about likelihood functions from nuclear physics

TL;DR

This paper investigates the appropriateness of the normal distribution for likelihood functions in nuclear physics data analysis, finding that a Student t distribution better models measurement disagreements and outliers.

Contribution

It demonstrates that a Student t distribution with an estimated degree of freedom provides a more accurate likelihood model for nuclear physics measurements than the traditional normal distribution.

Findings

Data variations exceed quoted uncertainties.

Student t distribution with degree of freedom ~2.6 fits data well.

Long-tailed t-distribution handles outliers effectively.

Abstract

Least-squares data analysis is based on the assumption that the normal (Gaussian) distribution appropriately characterizes the likelihood, that is, the conditional probability of each measurement d, given a measured quantity y, p(d | y). On the other hand, there is ample evidence in nuclear physics of significant disagreements among measurements, which are inconsistent with the normal distribution, given their stated uncertainties. In this study the histories of 99 measurements of the lifetimes of five elementary particles are examined to determine what can be inferred about the distribution of their values relative to their stated uncertainties. Taken as a whole, the variations in the data are somewhat larger than their quoted uncertainties would indicate. These data strongly support using a Student t distribution for the likelihood function instead of a normal. The most probable value for the order of the t distribution is 2.6 +/- 0.9. It is shown that analyses based on long-tailed t-distribution likelihoods gracefully cope with outlying data.