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22.4 Multivariate Gaussian Process Distribution

22.4.1 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.

22.4.2 Sampling Statement

y ~ multi_gp(Sigma, w)

Increment target log probability density with multi_gp_lpdf( y | Sigma, w) dropping constant additive terms.

22.4.3 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