Procedure: Transformed outputs¶
Description and Background¶
It is sometimes appropriate to build an emulator of some transformation of the simulator output of interest, rather than the output itself. See the discussion page on the Gaussian assumption (DiscGaussianAssumption) for the background to using output transformations.
The emulator allows inference statements about the transformed output, and for instance can be used to conduct uncertainty analysis or sensitivity analysis of the transformed output. However, the interest lies in making such inferences and analyses on the original, untransformed output. The procedure explains how to construct these from a fully Bayesian emulator of the transformed output.
In the case of a Bayes linear emulator, entirely different methods are needed to make inferences about the original output.
Inputs¶
The input is a fully Bayesian emulator for the transformed simulator output.
We will use the following notation. For any given input configuration \(x\), let the original output be \(f(x)\), and let the transformed output be \(t(x) = g\{f(x)\}\), so that \(g\) denotes the transformation. The emulator therefore provides a probability distribution for \(t(x)\) at any or all values of \(x\). We suppose that the transformation is one-to-one and that the inverse transformation is \(g^{-1}\), i.e. \(f(x) = g^{-1}\{t(x)\}\).
Outputs¶
Outputs are any desired inferences about properties of the original output. For instance, if we let the inputs be random, denoting them now by \(X\), then uncertainty analysis of \(f(X)\) might include as one specific inference the expectation (with respect to the code uncertainty) of the uncertainty mean \(M = \textrm{E}[f(X)]\). In this case the property is \(M\) and the inference is the mean (interpreted as an estimate of \(M\)).
Procedure¶
The simplest procedure is to use simulation. The method of simulation based inference for emulators requires only a little modification. The method involves drawing random realisations from the emulator distribution, and then computing the property in question for each such realisation. The set of property values so derived is a sample from the (code uncertainty) distribution of that parameter. From this sample we compute the necessary inferences.
When we have a transformed output, we add one more step. We draw random realisations from the emulator for \(t(x)\), but we now apply the inverse transformation \(g^{-1}\) to every point on the realisation before computing the property value. This ensures that the parameter values are now a sample from the distribution of that property as defined for the original output.
Additional Comments¶
There are specific cases where we can do better than this. For some transformations we can derive the distribution of \(f(x)\) for any given \(x\) analytically, at least conditionally on hyperparameters. In some cases, we may even be able to derive uncertainty analysis or sensitivity analysis. This is an area for ongoing research.