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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_lupdf(y | Sigma, w).

### 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

real multi_gp_lupdf(matrix y | matrix Sigma, vector w)
The log of the multivariate GP density of matrix y given kernel matrix Sigma and inverses scales w dropping constant additive terms