25.5 Multivariate Gaussian process distribution, Cholesky parameterization
25.5.1 Probability density function
If K,N∈N, L∈RN×N is lower triangular and such that LL⊤ is positive definite kernel matrix (implying Ln,n>0 for n∈1:N), and w∈RK is a vector of positive inverse scales, then for y∈RK×N, MultiGPCholesky(y| L,w)=K∏i=1MultiNormal(yi|0,w−1iLL⊤), where yi is the ith 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 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