Integrated likelihood computation methods

Suppose a model has parameter $$\theta =(\psi , \lambda )$$ θ = ( ψ , λ ) , where $$\psi $$ ψ is the parameter of interest and $$\lambda $$ λ is a nuisance parameter. The integrated likelihood method eliminates $$\lambda $$ λ from the likelihood function $$L(\psi , \lambda )$$ L ( ψ , λ ) by integrating with respect to a weight function $$\pi (\lambda | \psi )$$ π ( λ | ψ ) . The resulting integrated likelihood function $$\bar{L}(\psi )$$ L ¯ ( ψ ) can be used for inference for $$\psi $$ ψ . However, the analytical form for the integrated likelihood is not always available. This paper discusses 12 different approaches to computing the integrated likelihood. Some methods were originally developed for other computation purposes and they are modified to fit in the integrated likelihood framework. Methods considered include direct numerical integration methods such as Monte Carlo integration method, importance sampling, Laplace method; marginal likelihood computation methods; and methods for computing the marginal posterior density. Simulation studies and real data example are presented to evaluate and compare these methods empirically.