  Monte Carlo maximum likelihood (MCML) provides an elegant approach to find
maximum likelihood estimators (MLEs) for latent variable models. However, MCML
algorithms are computationally expensive when the latent variables are
high-dimensional and correlated, as is the case for latent Gaussian random
field models. Latent Gaussian random field models are widely used, for example
in building flexible regression models and in the interpolation of spatially
dependent data in many research areas such as analyzing count data in disease
modeling and presence-absence satellite images of ice sheets. We propose a
computationally efficient MCML algorithm by using a projection-based approach
to reduce the dimensions of the random effects. We develop an iterative method
for finding an effective importance function; this is generally a challenging
problem and is crucial for the MCML algorithm to be computationally feasible.
We find that our method is applicable to both continuous (latent Gaussian
process) and discrete domain (latent Gaussian Markov random field) models. We
illustrate the application of our methods to challenging simulated and real
data examples for which maximum likelihood estimation would otherwise be very
challenging. Furthermore, we study an often overlooked challenge in MCML
approaches to latent variable models: practical issues in calculating standard
errors of the resulting estimates, and assessing whether resulting confidence
intervals provide nominal coverage. Our study therefore provides useful
insights into the details of implementing MCML algorithms for high-dimensional
latent variable models.
