Functional Uniform Priors for Nonlinear Modelling

TL;DR

This paper introduces a novel class of prior distributions for nonlinear models that are uniform in the space of functional shapes, ensuring invariance and adherence to the likelihood principle, with applications demonstrated in clinical dose-finding trials.

Contribution

It proposes functional uniform priors based on metric space uniformity, offering a new approach for nonlinear modeling that is invariant and principled.

Findings

Functional uniform priors are invariant to parametrization.

Application to nonlinear regression in clinical trials shows improved modeling.

Used for optimal Bayesian design in dose-finding studies.

Abstract

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.