PARAMETER FITTING FOR UNCERTAIN MODELS - MODELING UNCERTAINTY IN SMALL MODELS

In uncertainty analysis distributions are placed over parameters in the model which is to be analysed. These distributions are sometimes said to express uncertainty in the parameter values, conditional on the model being correct. This paper takes the view that distributions over model parameters should express the unconditional uncertainty in the functional relationships posited by the model. For small models, effective procedures are given to generate distributions over model parameters which account for this type of uncertainty. A model is regarded as a function from one observation space to another. Uncertainty distributions over the image space, conditional on various values from the domain space are assumed to be given, e.g. via expert judgement. The problem is to define a unique distribution over the parameter space of the model which best utilises these conditional distributions. Two solution concepts are distinguished. A 'classical' solution uses an analogue of the log likelihood ratio to define a unique distribution on the model's parameter space. The solution best fits the expert conditional distributions when the latter are projected onto the parameter space of the model. A Bayesian solution considers the conditional distributions as data in an updating scheme. These two solution concepts are interpreted as correctly capturing different legitimate senses of the word 'solution'. Data from recent applications in atmospheric dispersion modelling and dose-response modelling are discussed.