Definition of Term: Implausibility Measure

An implausibility measure is a function \(I(x)\) defined over the whole input space which, if large for a particular \(x\), suggests that there would be a substantial disparity between the simulator output \(f(x)\) and the observed data \(z\), were we to evaluate the model at \(\strut{x}\).

In the simplest case where \(f(x)\) represents a single output and \(z\) a single observation, the univariate implausibility would look like:

\[I^2(x) = \frac{ ({\rm E}[f(x)] - z )^2}{ {\rm Var}[{\rm E}[f(x)]-z] } = \frac{ ({\rm E}[f(x)] - z )^2}{{\rm Var}[f(x)] + {\rm Var}[d] + {\rm Var}[e]}\]

where \({\rm E}[f(x)]\) and \({\rm Var}[f(x)]\) are the emulator expectation and variance respectively; \(d\) is the model discrepancy, and \(e\) is the observational error. The second equality follows from the definition of the best input approach.

Several different implausibility measures can be defined in the case where the simulator produces multiple outputs.