Lessons about likelihood functions from nuclear physics

Abstract

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.

Not Normal: the uncertainties of scientific measurements

“…The results of this study agree with earlier research that also observed Student-t tails, but only looked at a handful of subatomic or astrophysics quantities up to z ∼ 5 − 10 [16,19,[56][57][58]. Unsurprisingly, the tails reported here are mostly heavier than those reported for repeated measurements made with the same instrument (ν ∼ 3−9) [59][60][61], which should be closer to Normal since they are not independent and share most systematic effects.…”

“…Modelling the heavy tails may help us understand the observed distributions. One way is to assume that the measurement values are normally distributed with standard deviation t that is unknown but which has a probability distribution f (t) [15,[17][18][19]77]. The measured value x is then expected to have a probability distribution…”

“…In order to generate Student-t distributions, f(t) must be a scaled inverse chi-squared (or Gamma) distribution in t 2 [19,77]. This works mathematically, but why would variations in σ for independent measurements have such a distribution?…”

“…To examine the accuracy of reported uncertainties, this paper reviews multiple published measurements of many different quantities, looking at the differences between measurements of each quantity normalized by their reported uncertainties. Previous similar studies [15][16][17][18][19][20] reported on only a few hundred to a few thousand measurements, mostly in subatomic physics. This study reports on how well multiple measurements of the same quantity agree, and hence what are reasonable expectations for the reproducibility of published scientific measurements.…”