## 25.5 Multivariate Gaussian process distribution, Cholesky parameterization

### 25.5.1 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.

### 25.5.2 Sampling statement

y ~ multi_gp_cholesky(L, w)

Increment target log probability density with multi_gp_cholesky_lupdf(y | L, w).
Available since 2.5

### 25.5.3 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
Available since 2.12

real multi_gp_cholesky_lupdf(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 dropping constant additive terms
Available since 2.25