  The key operation in Bayesian inference, is to compute high-dimensional
integrals. An old approximate technique is the Laplace method or approximation,
which dates back to Pierre- Simon Laplace (1774). This simple idea approximates
the integrand with a second order Taylor expansion around the mode and computes
the integral analytically. By developing a nested version of this classical
idea, combined with modern numerical techniques for sparse matrices, we obtain
the approach of Integrated Nested Laplace Approximations (INLA) to do
approximate Bayesian inference for latent Gaussian models (LGMs). LGMs
represent an important model-abstraction for Bayesian inference and include a
large proportion of the statistical models used today. In this review, we will
discuss the reasons for the success of the INLA-approach, the R-INLA package,
why it is so accurate, why the approximations are very quick to compute and why
LGMs make such a useful concept for Bayesian computing.
