Efficient Emulation of Smooth Functions for Bayesian Inference

Bayesian parameter estimation for computationally intensive models often relies on surrogate emulators to efficiently explore high-dimensional parameter spaces. We present an emulation framework that models the simulator output as an analytic polynomial expansion with independent terms and decreasing weights governed by a convergence parameter. By enforcing invariance under rotation and translation of the input space, the resulting expansion becomes formally equivalent to a Gaussian process emulator with a Gaussian correlation kernel.

A smoothness scale, characterized by a coherence parameter Λ, quantifies the number of training points required to reproduce the model output and estimate uncertainties. We introduce an accuracy metric—defined as the expected uncertainty averaged over the prior distribution—and derive analytic expressions for this metric using only Λ and the positions of training points. This enables optimal placement of training points prior to data collection, significantly outperforming standard methods such as random and Latin hypercube sampling.

We demonstrate the effectiveness of the approach through examples, and investigate how emulator accuracy depends on model dimensionality, number of training points, and the convergence parameter. Our method offers a systematic and computationally efficient strategy for emulator design in Bayesian workflows involving expensive simulations.