Merge pull request #785 from stan-dev/gp_computation_cost_warning

warn about the computation cost of GPs

src/bibtex/all.bib

@@ -1816,3 +1816,12 @@ @article{zhang_pathfinder:2022
   url     = {http://jmlr.org/papers/v23/21-0889.html}
 }
 
+@article{Riutort-Mayol:2023:HSGP,
+  title={Practical {Hilbert} space approximate {Bayesian} {Gaussian} processes for probabilistic programming},
+  author={Riutort-Mayol, Gabriel and B{\"u}rkner, Paul-Christian and Andersen, Michael R and Solin, Arno and Vehtari, Aki},
+  journal={Statistics and Computing},
+  volume={33},
+  number={1},
+  pages={17},
+  year={2023}
+}

src/stan-users-guide/gaussian-processes.qmd

@@ -44,6 +44,16 @@ Gaussian processes are  general, and by necessity this chapter
 only touches on some basic models.  For more information, see
 @RasmussenWilliams:2006.
 
+Note that fitting Gaussian processes as described below using exact
+inference by computing Cholesky of the covariance matrix scales
+cubicly with the size of data. Due to how Stan autodiff is
+implemented, Stan is also slower than Gaussian process specialized
+software. It is likely that Gaussian processes using exact inference
+by computing Cholesky of the covariance matrix with $N>1000$ are too
+slow for practical purposes in Stan. There are many approximations to
+speed-up Gaussian process computation, from which the basis function
+approaches for 1-3 dimensional $x$ are easiest to implement in Stan
+(see, e.g., @Riutort-Mayol:2023:HSGP).
 
 ## Gaussian process regression