Definition of Term: Emulator

An emulator is a statistical representation of a simulator. For any given configuration of input values for the simulator, the emulator provides a probabilistic prediction of one or more of the outputs that the simulator would produce if it were run at those inputs.

Furthermore, for any set of input configurations, the emulator will provide a joint probabilistic prediction of the corresponding set of simulator outputs.

Example: A simulator of a nuclear power station reactor requires inputs that specify the flow of gas through the reactor, and produces an output which is the steady state mean temperature of the reactor core. A simulator of this output would provide probabilistic predictions of the mean temperature (as output by the actual simulator) at any single configuration of gas flow inputs, or at any set of such configurations.

The probabilistic predictions may take one of two forms depending on the approach used to build the emulator. In the fully Bayesian approach, the predictions are complete probability distributions.

Example: In the previous example, the fully Bayesian approach would provide a complete probability distribution for the output from any single configuration of inputs. This would give the probability that the output (mean temperature) lies in any required range. In particular, it would also provide any desired summaries of the probability distribution, such as the mean or variance. For a given set of input configurations, it would produce a joint probability distribution for the corresponding set of outputs, and in particular would give means, variances and covariances.

In the Bayes linear approach, the emulator’s probabilistic specification of outputs comprises (adjusted) means, variances and covariances.

Example: The Bayes linear emulator in the above example would provide the (adjusted) mean and (adjusted) variance for the simulator output (mean temperature) from a given single configuration of gas flow inputs. When the emulator is used to predict a set of outputs from more than one input configuration, it will also provide (adjusted) covariances between each pair of outputs.

Strictly, these ‘adjusted’ means, variances and covariances have a somewhat different meaning from the means, variances and covariances in a fully Bayesian emulator. Nevertheless, they are in practice interpreted the same way - thus, the (adjusted) mean is a point estimate of the simulator output and the square-root of the (adjusted) variance is a measure of accuracy for that estimate in the sense of being a root-mean-square distance from the estimate to the true simulator output. In practice, we would drop the word ‘adjusted’ and simply call them means, variances and covariances, but the distinction can be important.