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22.5 Multivariate Gaussian Process Distribution, Cholesky parameterization

22.5.1 Probability Density Function

If $K,N \in \mathbb{N}$, $L \in \mathbb{R}^{N \times N}$ is lower triangular and such that $LL^{\top}$ is positive definite kernel matrix (implying $L_{n,n} > 0$ for $n \in 1{:}N$), and $w \in \mathbb{R}^{K}$ is a vector of positive inverse scales, then for $y \in \mathbb{R}^{K \times N}$, $$\text{MultiGPCholesky}(y \, | \ L,w) = \prod_{i=1}^{K} \text{MultiNormal}(y_i|0,w_i^{-1} LL^{\top}),$$ 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. If the model allows parameterization in terms of Cholesky factor of the kernel matrix, this distribution is also more efficient than $\text{MultiGP}()$. Note that this function does not take into account the mean prediction.

22.5.2 Sampling Statement

y ~ multi_gp_cholesky(L, w)

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

22.5.3 Stan Functions

real multi_gp_cholesky_lpdf(matrix y | matrix L, vector w)
The log of the multivariate GP density of matrix y given lower-triangular Cholesky factor of the kernel matrix L and inverses scales w