Definition of Term: Multivariate t-process

The univariate t-process is a probability distribution for a function, in which the joint distribution of any set of points on that function is multivariate t. In MUCM it arises in the fully Bayesian approach as the underlying distribution (after integrating out a variance hyperparameter) of an emulator for the output of a simulator, regarding the simulator output as a function \(f(x)\) of its input(s) \(x\); see the procedure for building a Gaussian process emulator for the core problem (ProcBuildCoreGP). The t-process generalises the Gaussian process (GP) in the same way that a t distribution generalises the normal (or Gaussian) distribution.

Most simulators in practice produce multiple outputs (e.g. {temperature, pressure, wind speed, …}) for any given input configuration. If the simulator has \(r\) outputs then \(f(x)\) is \(r\times 1\). In this context, an emulator may be based on the multivariate GP or a multivariate t-process.

Formally, the multivariate t-process is a probability model over functions with multivariate values. It is characterised by a degrees of freedom \(b\), a mean function \(m(\cdot) = \textrm{E}[f(\cdot)]\) and a covariance function \(v(\cdot,\cdot) = \textrm{Cov}[f(\cdot),f(\cdot)]\). Under this model, the function evaluated at a single input \(x\) has a multivariate t distribution with \(b\) degrees of freedom, where:

  • \(m(x)\) is the \(r \times 1\) mean vector of \(f(x)\) and
  • \(v(x,x)\) is the \(r\times r\) scale matrix of \(f(x)\).

Furthermore, if we stack the vectors \(f(x_1), f(x_2),\cdots,f(x_n)\) at an arbitrary set of \(n\) outputs \(D = (x_1,\cdots,x_n)\) into a vector of \(rn\) elements, then this also has a multivariate t distribution.

The multivariate t-process usually arises when we use a multivariate GP model with a separable covariance. We therefore shall assume unless specified otherwise that a multivariate t-process also has a separable covariance, that is

\[v(\cdot,\cdot) = \Sigma c(\cdot, \cdot)\]

where \(\Sigma\) is a covariance matrix between outputs and \(c(\cdot, \cdot)\) is a correlation function between input points.

With a covariance function of this form the multivariate t-process has an important property. If instead of stacking the output vectors into a long vector we form instead the \(r\times n\) matrix \(f(D)\) (following the conventions in the notation page) then \(f(D)\) has a matrix-variate t distribution with \(b\) degrees of freedom, mean matrix \(m(D)\), between-rows covariance matrix \(\Sigma\) and between-columns covariance matrix \(c(D,D)\).

References

Information on matrix variate distributions can be found in: