  Gaussian processes (GPs) are pervasive in functional data analysis, machine
learning, and spatial statistics for modeling complex dependencies. Modern
scientific data sets are typically heterogeneous and often contain multiple
known discrete subgroups of samples. For example, in genomics applications
samples may be grouped according to tissue type or drug exposure. In the
modeling process it is desirable to leverage the similarity among groups while
accounting for differences between them. While a substantial literature exists
for GPs over Euclidean domains $\mathbb{R}^p$, GPs on domains suitable for
multi-group data remain less explored. Here, we develop a multi-group Gaussian
process (MGGP), which we define on $\mathbb{R}^p\times \mathscr{C}$, where
$\mathscr{C}$ is a finite set representing the group label. We provide general
methods to construct valid (positive definite) covariance functions on this
domain, and we describe algorithms for inference, estimation, and prediction.
We perform simulation experiments and apply MGGP to gene expression data to
illustrate the behavior and advantages of the MGGP in the joint modeling of
continuous and categorical variables.
