Definition of Term: T-process

A t-process is a probability model for an unknown function.

If a function f has argument \(x\), then the value of the function at that argument is \(f(x)\). A probability model for such a function must provide a probability distribution for \(f(x)\), for any possible argument value \(x\). Furthermore, if we consider a set of possible values for the argument, which we can denote by \(x_1, x_2, ..., x_n\), then the probability model must provide a joint probability distribution for the corresponding function values \(f(x_1), f(x_2), ..., f(x_n)\).

In MUCM methods, an emulator is (at least in the fully Bayesian approach) a probability model for the corresponding simulator. The simulator is regarded as a function, with the simulator inputs comprising the function’s argument and the simulator output(s) comprising the function value.

A t-process is a particular probability model in which the distribution for a single function value is a t distribution (also often called a Student-t distribution), and the joint distribution of a set of function values is multivariate t. The t-process is related to the Gaussian process (GP) in the same way that a univariate or multivariate t distribution is related to a univariate or multivariate normal (or Gaussian) distribution. That is, we can think of a t-process as a GP with an uncertain (or random) scaling parameter in its covariance function. Normal and t distributions play a central role in statistical practice by virtue of their mathematical simplicity, and for similar reasons the GP and t-process play a central role in (fully Bayesian) MUCM methods.

A t-process is identified by its mean function, its covariance function and a degrees of freedom. If the degrees of freedom is sufficiently large, which will in practice always be the case in MUCM applications, the mean function is \(\text{E}[f(x)]\), regarded as a function of \(x\), and the covariance function is \(\text{Cov}[f(x_1), f(x_2)]\), regarded as a function of both \(x_1\) and \(x_2\). Note in particular that the variance of \(f(x)\) is the value of the covariance function when both \(x_1\) and \(x_2\) are equal to \(x\).