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When using R package tmbstan for Bayesian inference, the built-in featureLaplace approximation to the marginal likelihood with random effects integratedout can be switched on and off. There exists no guideline on whether Laplaceapproximation should be used to achieve better efficiency especially when thestatistical model for estimating selection is complicated. To answer thisquestion, we conducted simulation studies under different scenarios with astate-space model employing a VAR(1) state equation. We found that turning onLaplace approximation in tmbstan would probably lower the computationalefficiency, and only when there is a good amount of data, both tmbstan with andwithout Laplace approximation are worth trying since in this case, Laplaceapproximation is more likely to be accurate and may also lead to slightlyhigher computational efficiency. The transition parameters and scale parametersin a VAR(1) process are hard to be estimated accurately and increasing thesample size at each time point do not help in the estimation, only more timepoints in the data contain more information on these parameters and make thelikelihood dominate the posterior likelihood, thus lead to accurate estimatesfor them.

Bayesian inference with tmbstan for a state-space model with VAR(1) state equation