  This paper considers the topic of finding prior distributions when a major
component of the statistical model depends on a nonlinear function. Using
results on how to construct uniform distributions in general metric spaces, we
propose a prior distribution that is uniform in the space of functional shapes
of the underlying nonlinear function and then back-transform to obtain a prior
distribution for the original model parameters. The primary application
considered in this article is nonlinear regression, but the idea might be of
interest beyond this case. For nonlinear regression the so constructed priors
have the advantage that they are parametrization invariant and do not violate
the likelihood principle, as opposed to uniform distributions on the parameters
or the Jeffrey's prior, respectively. The utility of the proposed priors is
demonstrated in the context of nonlinear regression modelling in clinical
dose-finding trials, through a real data example and simulation. In addition
the proposed priors are used for calculation of an optimal Bayesian design.
