Multivariate Normal distribution and Cholesky decomposition in Stan
This post provides an example of simulating data in a Multivariate Normal distribution with given parameters, and estimating the parameters based on the simulated data via Cholesky decomposition in stan. Multivariate Normal distribution is a commonly used distribution in various regression models and machine learning tasks. It generalizes the Normal distribution into multidimensional space. Its PDF can be expressed as:
where mu is a vector of k elements for the location parameters and Sigma is the Variance-Covariance matrix. The |Sigma| calculates the absolute determinant of Sigma in the equation above.
Cholesky decomposition is used to decompose a positive-definite matrix into the product of a lower triangular matrix and its transpose. In our case, we will decompose the correlation matrix (R) into the product of two triangular matrices (L*L’). Note: even though we can directly perform Cholesky decomposition on the Variance-Covariance matrix (Sigma), it is not recommended, since we may fail to estimate the standard deviations (sigma) of each variable.
