Explore the Formulation of Laplacian
For the Laplacian approach, the Hessian of 
is directly calculated either analytically or numerically (through some finite-difference schemes), and 
is chosen to be the conditional mode 
obtained with population parameter 
, 
, 
and 
, i = 1, 2,..., NSUB. Since the approximated marginal PDF for this approach also depends on the conditional modes, it is implemented using the same basic structure as the Laplacian approximation based FOCE-ELS. In other words, it involves a single top-level optimization:
where the top-level objective function, 
, is given by
For each evaluation of the objective function, one needs to do NSUB inner optimization problems by minimizing 
(discussed earlier) with population parameters set to be their corresponding estimates obtained at the previous step to evaluate the objective function.