Optimally Emulating Smooth Functions for Bayesian Analyses

Abstract

For Bayesian analyses aimed at constraining the parameter space of computationally expensive models by comparison with heterogeneous data sets, model emulators play a pivotal role. Model emulators enable searches through high-dimensional parameter spaces, searches which might otherwise require millions of full model runs. Here, an emulation strategy is presented based on the assumption that the model output can be well described by a polynomial expansion with all terms being independent of one another and with higher order terms being given decreasing weights according to a convergence parameter. It is found that if the expansion's form is constrained to being invariant to rotation and translation of the parameter space that the polynomial form becomes equivalent to that of a Gaussian process emulator using a correlation kernel with a Gaussian form. In the limit that the function is smooth, quantified by a convergence parameter, $Λ$, it is found that rather few training points are required to reproduce the full model and provide a good estimate of its uncertainty. An accuracy metric is defined based on an average of the expected uncertainty over the prior before training data are actually gathered. Analytic expressions for the expected accuracy are provided based solely on an expected value of $Λ$ and the location of the training points. This enables a tenable optimized placement of training points, which is shown to be significantly superior to random placement or that from Latin hyper-cube sampling. Examples of optimized placement are provided, and the accuracy is studied as a function of various variables, such as the number of model parameters, the number of training points, and the convergence parameter.