Integrated nested Laplace approximations
Integrated nested Laplace approximations (INLA) is a method for approximate Bayesian inference based on Laplace's method. It is designed for a class of models called latent Gaussian models (LGMs), for which it can be a fast and accurate alternative for Markov chain Monte Carlo methods to compute posterior marginal distributions. Due to its relative speed even with large data sets for certain problems and models, INLA has been a popular inference method in applied statistics, in particular spatial statistics, ecology, and epidemiology. It is also possible to combine INLA with a finite element method solution of a stochastic partial differential equation to study e.g. spatial point processes and species distribution models. The INLA method is implemented in the R-INLA R package. Latent Gaussian models 
Let  denote the response variable (that is, the observations) which belongs to an exponential family, with the mean  (of ) being linked to a linear predictor  via an appropriate link function. The linear predictor can take the form of a (Bayesian) additive model. All latent effects (the linear predictor, the intercept, coefficients of possible covariates, and so on) are collectively denoted by the vector . The hyperparameters of the model are denoted by . As per Bayesian statistics,  and  are random variables with prior distributions. The observations are assumed to be conditionally independent given  and :

where  is the set of indices for observed elements of  (some elements may be unobserved, and for these INLA computes a posterior predictive distribution). Note that the linear predictor  is part of . For the model to be a latent Gaussian model, it is assumed that  is a Gaussian Markov Random Field (GMRF) (that is, a multivariate Gaussian with additional conditional independence properties) with probability density 

where  is a -dependent sparse precision matrix and  is its determinant. The precision matrix is sparse due to the GMRF assumption. The prior distribution  for the hyperparameters need not be Gaussian. However, the number of hyperparameters, , is assumed to be small (say, less than 15). Approximate Bayesian inference with INLA 
In Bayesian inference, one wants to solve for the posterior distribution of the latent variables  and . Applying Bayes' theorem

the joint posterior distribution of  and  is given by

Obtaining the exact posterior is generally a very difficult problem. In INLA, the main aim is to approximate the posterior marginals

where . A key idea of INLA is to construct nested approximations given by

where  is an approximated posterior density. The approximation to the marginal density  is obtained in a nested fashion by first approximating  and , and then numerically integrating out  as

where the summation is over the values of , with integration weights given by . The approximation of  is computed by numerically integrating  out from . To get the approximate distribution , one can use the relation

as the starting point. Then  is obtained at a specific value of the hyperparameters  with Laplace's approximation

where  is the  Gaussian approximation to  whose mode at a given  is . The mode can be found numerically for example with the Newton-Raphson method. The trick in the Laplace approximation above is the fact that the Gaussian approximation is applied on the full conditional of  in the denominator since it is usually close to a Gaussian due to the GMRF property of . Applying the approximation here improves the accuracy of the method, since the posterior  itself need not be close to a Gaussian, and so the Gaussian approximation is not directly applied on . The second important property of a GMRF, the sparsity of the precision matrix , is required for efficient computation of  for each value . Obtaining the approximate distribution  is more involved, and the INLA method provides three options for this: Gaussian approximation, Laplace approximation, or the simplified Laplace approximation. For the numerical integration to obtain , also three options are available: grid search, central composite design, or empirical Bayes. References

Further reading 
 

Computational statistics
Bayesian inference