in *Bayesian Inference and Maximum Entropy Methods in Science and Engineering*, K. H. Knuth et al., eds., *AIP Conf. Proc.* **954**, pp. 458-467 (AIP, Melville, 2007), DOI: 10.1063/1.2821298, arXiv: 0712.0021

## Lessons about likelihood functions from nuclear physics

Kenneth M. Hanson

*Los Alamos National Laboratory*
### 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.

**Keywords:** likelihood, Student t distribution, long-tailed likelihood functions, systematic uncertainty, inconsistent data, outliers, robust analysis, least-squares analysis

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