Editing Computer experiment (section)

==Computer simulation modeling==
Modeling of computer experiments typically uses a Bayesian framework. [[Bayesian statistics]] is an interpretation of the field of [[statistics]] where all evidence about the true state of the world is explicitly expressed in the form of [[probabilities]].  In the realm of computer experiments, the Bayesian interpretation would imply we must form a [[prior distribution]] that represents our prior belief on the structure of the computer model.  The use of this philosophy for computer experiments started in the 1980s and is nicely summarized by Sacks et al. (1989) [https://web.archive.org/web/20170918022130/https://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ss%2F1177012413].  While the Bayesian approach is widely used, [[frequentist]] approaches have been recently discussed [http://www2.isye.gatech.edu/~jeffwu/publications/calibration-may1.pdf].

The basic idea of this framework is to model the computer simulation as an unknown function of a set of inputs.  The computer simulation is implemented as a piece of computer code that can be evaluated to produce a collection of outputs.  Examples of inputs to these simulations are coefficients in the underlying model, [[initial conditions]] and [[Forcing function (differential equations)|forcing functions]].  It is natural to see the simulation as a deterministic function that maps these ''inputs'' into a collection of ''outputs''.  On the basis of seeing our simulator this way, it is common to refer to the collection of inputs as  <math>x</math>, the computer simulation itself as <math>f</math>, and the resulting output as <math>f(x)</math>.  Both <math>x</math> and <math>f(x)</math> are vector quantities, and they can be very large collections of values, often indexed by space, or by time, or by both space and time.

Although <math>f(\cdot)</math> is known in principle, in practice this is not the case.  Many simulators comprise tens of thousands of lines of high-level computer code, which is not accessible to intuition.  For some simulations, such as climate models, evaluation of the output for a single set of inputs can require millions of computer hours [http://amstat.tandfonline.com/doi/abs/10.1198/TECH.2009.0015#.UbixC_nFWHQ].

===Gaussian process prior===
The typical model for a computer code output is a Gaussian process.  For notational simplicity, assume <math> f(x) </math> is a scalar.  Owing to the Bayesian framework, we fix our belief that the function <math>f</math> follows a [[Gaussian process]],
<math>f \sim \operatorname{GP}(m(\cdot),C(\cdot,\cdot)),</math>
where <math> m</math> is the mean function and <math> C </math> is the covariance function.  Popular mean functions are low order polynomials and a popular [[covariance function]] is [[Matern covariance]], which includes both the exponential (<math> \nu = 1/2 </math>) and Gaussian covariances (as <math> \nu \rightarrow \infty </math>).