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Appendix D. Priors and posteriors for variance parameters.

The joint prior distribution on *σ*_{τ} and *σ*_{ε} used in the main text, , results in a marginally uniform distribution for both *σ*_{ε} and *σ*_{τ} but is strongly informative in their ratio. It is used to represent the prior belief that both process and measurement errors are present, and would in real-world applications be replaced with other prior information regarding the absolute value or relative magnitude of process and measurement errors (as discussed further in the main text). To visualize the impact of this prior on model results, we show the joint prior distribution with the posterior mean for each replicate superimposed (Fig. D1–D2). This illustrates that the model is generally able to estimate the absolute scale of errors, and correctly identifies that process and measurement errors have equal magnitude in the ‘matched function’ setting. However, when the specified or prior function does not match the function generating the data, the conventional and GP models compensate by estimating that process errors have a greater magnitude that measurement errors. This is most obviously the case for the ‘depensatory’ function, where process errors are estimated to be much larger than measurement errors (and where the posterior mean for many replicates has a prior probability of <0.001 relative to the maximum prior probability). This difference between prior and posterior means is largely absent for the high variability scenarios, when the posterior mean is again close to the prior mean. This confirms that the joint prior is particularly informative when the data have little alternative information regarding the relative magnitude of process and measurement errors. In these cases, a real-world application of the conventional of Gaussian process models would likely require prior information regarding one or the other variance, i.e., by using sampling theory to estimate the magnitude of measurement errors.

Fig. D1. Log–log plot comparing the joint prior for process error variability *σ*_{ε} and measurement error variability *σ*_{τ} (using contour lines: black line: 50% of prior maximum; long-dashed line: 10% of prior maximum; dashed line: 1% of prior maximum; dotted line: 0.1% of prior maximum) and the posterior mean estimates for each replicate (dots, where blue: conventional; red: Gaussian process) in the absolute index scenario.