  Fitting models to data using Bayesian inference is quite common, but when
each point in parameter space gives a curve, fitting the curve to a data set
requires new nuisance parameters, which specify the metric embedding the
one-dimensional curve into the higher-dimensional space occupied by the data. A
generic formalism for curve fitting in the context of Bayesian inference is
developed which shows how the aforementioned metric arises. The result is a
natural generalization of previous works, and is compared to oft-used
frequentist approaches and similar Bayesian techniques.
