Integrated nested Laplace approximations (INLA) is a method for approximate Bayesian inference based on Laplace's method.
[1] 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.
[2][3][4] 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.
[5][6][7] 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.
[8][9] The INLA method is implemented in the R-INLA R package.
denote the response variable (that is, the observations) which belongs to an exponential family, with the mean
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
are random variables with prior distributions.
The observations are assumed to be conditionally independent given
is the set of indices for observed elements of
(some elements may be unobserved, and for these INLA computes a posterior predictive distribution).
is a Gaussian Markov Random Field (GMRF)[1] (that is, a multivariate Gaussian with additional conditional independence properties) with probability density
The precision matrix is sparse due to the GMRF assumption.
In Bayesian inference, one wants to solve for the posterior distribution of the latent variables
the joint posterior distribution of
Obtaining the exact posterior is generally a very difficult problem.
In INLA, the main aim is to approximate the posterior marginals
A key idea of INLA is to construct nested approximations given by
is obtained in a nested fashion by first approximating
is obtained at a specific value of the hyperparameters
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
The second important property of a GMRF, the sparsity of the precision matrix
[1] 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.
[1] For the numerical integration to obtain
, also three options are available: grid search, central composite design, or empirical Bayes.