Inference for deterministic simulation models: The Bayesian melding approach

Deterministic simulation models are used in many areas of science, engineering, and policy making. Typically, these are complex models that attempt to capture underlying mechanisms in considerable detail, and they have many user-specified inputs. The inputs are often specified by some form of trial-and-error approach in which plausible values are postulated the corresponding outputs inspected, and the inputs modified until plausible outputs are obtained. Here we address the issue of more formal inference for such models. A probabilistic approach, called Bayesian synthesis, was shown to suffer from the Borel paradox, according to which the results can depend on the parameterization of the model. We propose a modified approach, called Bayesian melding, which takes into full account information and uncertainty about both inputs and outputs to the model, while avoiding the Borel paradox. This is done by recognizing the existence of two priors, one implicit and one explicit, on each input and output; these are combined via logarithmic pooling. Bayesian melding is then standard Bayesian inference with the pooled prior on inputs, and is implemented here by posterior simulation using the sampling-importance-resampling (SIR) algorithm. We develop this initially for invertible models, and then extend it to the more difficult and more common case of noninvertible models. We illustrate the methodology using a number of examples. Simulation studies show that the method outperforms a simpler Bayesian approach in terms of mean squared error. A number of open research problems are discussed.