Information Matrix Splitting

Efficient statistical estimates via the maximum likelihood method requires the observed information, the negative of the Hessian of the underlying log-likelihood function. Computing the observed information is computationally prohibitive for high-throughput biological data, therefore, the expected information matrix---the Fisher information matrix---is often preferred due to its simplicity. In this paper, we prove that the average of the observed and the Fisher information of restricted/residual log-likelihood functions for linear mixed models can be split into two matrices. The expectation of one part is the Fisher information matrix but enjoys a simper formula than the Fisher information matrix. The other part which involves a lot of computations is a random zero matrix and thus is negligible. Leveraging such a splitting can simplify evaluation of the approximate Hessian of a log-likelihood function.