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## 22.3 Multivariate normal distribution, Cholesky parameterization

### 22.3.1 Probability density function

If $$K \in \mathbb{N}$$, $$\mu \in \mathbb{R}^K$$, and $$L \in \mathbb{R}^{K \times K}$$ is lower triangular and such that $$LL^{\top}$$ is positive definite, then for $$y \in \mathbb{R}^K$$, $\text{MultiNormalCholesky}(y|\mu,L) = \text{MultiNormal}(y|\mu,LL^{\top}).$ If $$L$$ is lower triangular and $$LL^{top}$$ is a $$K \times K$$ positive definite matrix, then $$L_{k,k}$$ must be strictly positive for $$k \in 1{:}K$$. If an $$L$$ is provided that is not the Cholesky factor of a positive-definite matrix, the probability functions will raise errors.

### 22.3.2 Sampling statement

y ~ multi_normal_cholesky(mu, L)

Increment target log probability density with multi_normal_cholesky_lupdf(y | mu, L).

### 22.3.3 Stan functions

real multi_normal_cholesky_lpdf(vectors y | vectors mu, matrix L)
The log of the multivariate normal density of vector(s) y given location vector(s) mu and lower-triangular Cholesky factor of the covariance matrix L

real multi_normal_cholesky_lupdf(vectors y | vectors mu, matrix L)
The log of the multivariate normal density of vector(s) y given location vector(s) mu and lower-triangular Cholesky factor of the covariance matrix L dropping constant additive terms

real multi_normal_cholesky_lpdf(vectors y | row_vectors mu, matrix L)
The log of the multivariate normal density of vector(s) y given location row vector(s) mu and lower-triangular Cholesky factor of the covariance matrix L

real multi_normal_cholesky_lupdf(vectors y | row_vectors mu, matrix L)
The log of the multivariate normal density of vector(s) y given location row vector(s) mu and lower-triangular Cholesky factor of the covariance matrix L dropping constant additive terms

real multi_normal_cholesky_lpdf(row_vectors y | vectors mu, matrix L)
The log of the multivariate normal density of row vector(s) y given location vector(s) mu and lower-triangular Cholesky factor of the covariance matrix L

real multi_normal_cholesky_lupdf(row_vectors y | vectors mu, matrix L)
The log of the multivariate normal density of row vector(s) y given location vector(s) mu and lower-triangular Cholesky factor of the covariance matrix L dropping constant additive terms

real multi_normal_cholesky_lpdf(row_vectors y | row_vectors mu, matrix L)
The log of the multivariate normal density of row vector(s) y given location row vector(s) mu and lower-triangular Cholesky factor of the covariance matrix L

real multi_normal_cholesky_lupdf(row_vectors y | row_vectors mu, matrix L)
The log of the multivariate normal density of row vector(s) y given location row vector(s) mu and lower-triangular Cholesky factor of the covariance matrix L dropping constant additive terms

vector multi_normal_cholesky_rng(vector mu, matrix L)
Generate a multivariate normal variate with location mu and lower-triangular Cholesky factor of the covariance matrix L; may only be used in transformed data and generated quantities blocks

vector multi_normal_cholesky_rng(row_vector mu, matrix L)
Generate a multivariate normal variate with location mu and lower-triangular Cholesky factor of the covariance matrix L; may only be used in transformed data and generated quantities blocks

vectors multi_normal_cholesky_rng(vectors mu, matrix L)
Generate an array of multivariate normal variates with locations mu and lower-triangular Cholesky factor of the covariance matrix L; may only be used in transformed data and generated quantities blocks

vectors multi_normal_cholesky_rng(row_vectors mu, matrix L)
Generate an array of multivariate normal variates with locations mu and lower-triangular Cholesky factor of the covariance matrix L; may only be used in transformed data and generated quantities blocks