  We present a unified view of likelihood based Gaussian progress regression
for simulation experiments exhibiting input-dependent noise. Replication plays
an important role in that context, however previous methods leveraging
replicates have either ignored the computational savings that come from such
design, or have short-cut full likelihood-based inference to remain tractable.
Starting with homoskedastic processes, we show how multiple applications of a
well-known Woodbury identity facilitate inference for all parameters under the
likelihood (without approximation), bypassing the typical full-data sized
calculations. We then borrow a latent-variable idea from machine learning to
address heteroskedasticity, adapting it to work within the same thrifty
inferential framework, thereby simultaneously leveraging the computational and
statistical efficiency of designs with replication. The result is an
inferential scheme that can be characterized as single objective function,
complete with closed form derivatives, for rapid library-based optimization.
Illustrations are provided, including real-world simulation experiments from
manufacturing and the management of epidemics.
