21.4 Multivariate Gaussian Process Distribution
21.4.1 Probability Density Function
If \(K,N \in \mathbb{N}\), \(\Sigma \in \mathbb{R}^{N \times N}\) is symmetric, positive definite kernel matrix and \(w \in \mathbb{R}^{K}\) is a vector of positive inverse scales, then for \(y \in \mathbb{R}^{K \times N}\), \[ \text{MultiGP}(y|\Sigma,w) = \prod_{i=1}^{K} \text{MultiNormal}(y_i|0,w_i^{-1} \Sigma), \] where \(y_i\) is the \(i\)th row of \(y\). This is used to efficiently handle Gaussian Processes with multi-variate outputs where only the output dimensions share a kernel function but vary based on their scale. Note that this function does not take into account the mean prediction.
21.4.2 Sampling Statement
y ~ multi_gp(Sigma, w)
Increment target log probability density with multi_gp_lpdf(y | Sigma, w)
dropping constant additive terms.
21.4.3 Stan Functions
real multi_gp_lpdf(matrix y | matrix Sigma, vector w)
The log of the multivariate GP density of matrix y given kernel matrix
Sigma and inverses scales w