Functional Uniform Priors for Nonlinear Modelling

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.

💡 Research Summary

The paper addresses a long‑standing problem in Bayesian nonlinear modelling: how to choose a prior when the model’s core component is a nonlinear function of the parameters. Conventional choices—flat priors on the parameters or Jeffreys’ priors—are unsatisfactory. Flat priors are not invariant under re‑parameterisation, so the prior can change dramatically if the same model is expressed with a different set of parameters. Jeffreys’ priors, while invariant, depend on the Fisher information matrix and therefore violate the likelihood principle because they incorporate information about the sampling distribution beyond the data at hand.

To overcome these drawbacks the authors propose a “functional uniform prior”. The key idea is to move the uniformity requirement from the parameter space to the space of functional shapes that the nonlinear mapping can take. They formalise the space of functions ( \mathcal{F} = { f(\cdot;\theta) : \theta \in \Theta } ) as a metric space equipped with a distance ( d(f_1,f_2) ) (e.g., an (L_2) norm or supremum norm). In any metric space a uniform distribution can be defined by the limiting behaviour of the volume of balls: a probability measure ( \Pi ) is uniform if for any centre (f) and radius (r) the probability of the ball (B(f,r)) is proportional to its volume, and this proportionality holds as (r) grows without bound.

Applying this construction to (\mathcal{F}) yields a prior that is uniform over all admissible functional shapes. The prior on the original parameters (\theta) is then obtained by the change‑of‑variables formula, i.e. by multiplying the functional uniform density by the Jacobian determinant of the transformation (\theta \mapsto f(\cdot;\theta)). Because the Jacobian accounts for the local stretching of the mapping, the resulting prior on (\theta) is invariant under any smooth re‑parameterisation ( \phi = g(\theta) ). Consequently the prior respects the likelihood principle: it does not depend on the sampling distribution beyond the data, and it does not introduce hidden information about the Fisher information.

The authors illustrate the methodology with several dose‑response models commonly used in phase‑I clinical trials, such as the Emax, logistic, and log‑normal models. Using a real dose‑finding dataset, they compare three priors: (i) a naïve flat prior on the parameters, (ii) Jeffreys’ prior, and (iii) the proposed functional uniform prior. Posterior summaries show that the functional uniform prior yields smaller mean‑squared error for the estimated dose‑response curve, tighter yet realistic credible intervals, and markedly reduced sensitivity to the choice of parameterisation, especially when the sample size is modest (n < 30). A systematic simulation study confirms these findings across a range of true dose‑response shapes, noise levels, and sample sizes.

Beyond inference, the paper demonstrates how the functional uniform prior can be incorporated into Bayesian optimal design. Using the D‑optimality criterion and the average prediction error as performance measures, the authors show that designs generated under the functional uniform prior allocate experimental points more evenly across the dose range, leading to up to a 12 % reduction in expected prediction error compared with designs based on flat priors. This improvement is most pronounced when the underlying dose‑response curve is highly nonlinear, underscoring the practical advantage of a prior that reflects genuine ignorance about functional shape rather than about arbitrary parameter values.

The discussion acknowledges that the choice of metric on (\mathcal{F}) influences the induced prior on (\theta); however, the authors argue that any reasonable metric that respects the topology of the function space will produce priors with the same desirable invariance properties. They also point out that extending the approach to multivariate function spaces, mixed‑effects nonlinear models, or dynamical systems will require additional theoretical work, particularly concerning the computation of the Jacobian in high dimensions.

In summary, the paper introduces a principled, mathematically rigorous way to construct priors that are uniform over the space of possible nonlinear functions rather than over the parameter space. This functional uniform prior is invariant to re‑parameterisation, complies with the likelihood principle, and can be efficiently implemented in both inference and design problems. The empirical results in dose‑finding trials demonstrate its superior performance relative to traditional flat and Jeffreys’ priors, suggesting that the approach could become a valuable tool for a broad range of Bayesian nonlinear modelling applications.