  There are several assumptions made in a standard $\chi^2$ analysis of data,
including the frequent assumption that the likelihood function is well
approximated by a multivariate Gaussian distribution. This article briefly
reviews the standard approach and describes how Bayesian inference can be used
to go beyond the assumption that the likelihood is Gaussian. Two separate types
of analysis relevant to nuclear physics are used as test cases. The first is
the determination of the equation of state of dense matter from neutron star
mass and radius data. The second is the use of theoretical nuclear mass models
to fit currently available data and predict the value of masses which have not
yet been measured. For the problem of predicting nuclear masses, it is
demonstrated that approximating the likelihood function with a Gaussian can
produce biased predictions of unmeasured masses. Finally, the lessons learned
from these fitting problems are used to propose a method for improving
constraints on the nuclear symmetry energy.
