Gearing Gaussian process modeling and sequential design towards stochastic simulators

Accurately reproducing real-world dynamics often requires stochastic simulators, particularly in fields like epidemiology, operations research, and hyperparameter tuning. In these contexts it becomes important to distinguish between aleatoric uncertainty - arising from noise in observations, from epistemic uncertainty - stemming from uncertainty in the model. The former is sometimes called intrinsic uncertainty while the latter is referred to as extrinsic uncertainty, see e.g., Ankenman et al. (2010). Gaussian process (GP) based surrogate methods (see, e.g., Rasmussen and Williams (2006); Gramacy (2020)) can be easily adapted from deterministic to noisy settings while maintaining strong predictive power, computational efficiency, and analytical tractability. Even in the deterministic setup, it is common to add a small diagonal nugget (also known as a jitter) term to the covariance matrix of the GP equations to ease its numerical inversion. It is also interpreted as a regularization term, especially in the reproducing kernel Hilbert space (RKHS) context, see, e.g., Kanagawa et al. (2018). This can be contrasted to the use of pseudo-inverses, which reverts to interpolation, see for instance the discussion by Mohammadi et al. (2016). Here we will prefer the term noise variance to relate it to intrinsic uncertainty, and also because the nugget effect has a different meaning in the kriging literature (see e.g., Roustant et al. (2012)).