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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.^[10]

Latent Gaussian models

Let ${\displaystyle {\boldsymbol {y}}=(y_{1},\dots ,y_{n})}$ denote the response variable (that is, the observations) which belongs to an exponential family, with the mean ${\displaystyle \mu _{i}}$ (of ${\displaystyle y_{i}}$) being linked to a linear predictor ${\displaystyle \eta _{i}}$ 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 ${\displaystyle {\boldsymbol {x}}}$. The hyperparameters of the model are denoted by ${\displaystyle {\boldsymbol {\theta }}}$. As per Bayesian statistics, ${\displaystyle {\boldsymbol {x}}}$ and ${\displaystyle {\boldsymbol {\theta }}}$ are random variables with prior distributions.

The observations are assumed to be conditionally independent given ${\displaystyle {\boldsymbol {x}}}$ and ${\displaystyle {\boldsymbol {\theta }}}$: $${\displaystyle \pi ({\boldsymbol {y}}|{\boldsymbol {x}},{\boldsymbol {\theta }})=\prod _{i\in {\mathcal {I}}}\pi (y_{i}|\eta _{i},{\boldsymbol {\theta }}),}$$ where ${\displaystyle {\mathcal {I}}}$ is the set of indices for observed elements of ${\displaystyle {\boldsymbol {y}}}$ (some elements may be unobserved, and for these INLA computes a posterior predictive distribution). Note that the linear predictor ${\displaystyle {\boldsymbol {\eta }}}$ is part of ${\displaystyle {\boldsymbol {x}}}$.

For the model to be a latent Gaussian model, it is assumed that ${\displaystyle {\boldsymbol {x}}|{\boldsymbol {\theta }}}$ is a Gaussian Markov Random Field (GMRF)^[1] (that is, a multivariate Gaussian with additional conditional independence properties) with probability density $${\displaystyle \pi ({\boldsymbol {x}}|{\boldsymbol {\theta }})\propto \left|{\boldsymbol {Q_{\theta }}}\right|^{1/2}\exp \left(-{\frac {1}{2}}{\boldsymbol {x}}^{T}{\boldsymbol {Q_{\theta }}}{\boldsymbol {x}}\right),}$$ where ${\displaystyle {\boldsymbol {Q_{\theta }}}}$ is a ${\displaystyle {\boldsymbol {\theta }}}$-dependent sparse precision matrix and ${\displaystyle \left|{\boldsymbol {Q_{\theta }}}\right|}$ is its determinant. The precision matrix is sparse due to the GMRF assumption. The prior distribution ${\displaystyle \pi ({\boldsymbol {\theta }})}$ for the hyperparameters need not be Gaussian. However, the number of hyperparameters, ${\displaystyle m=\mathrm {dim} ({\boldsymbol {\theta }})}$, 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 ${\displaystyle {\boldsymbol {x}}}$ and ${\displaystyle {\boldsymbol {\theta }}}$. Applying Bayes' theorem $${\displaystyle \pi ({\boldsymbol {x}},{\boldsymbol {\theta }}|{\boldsymbol {y}})={\frac {\pi ({\boldsymbol {y}}|{\boldsymbol {x}},{\boldsymbol {\theta }})\pi ({\boldsymbol {x}}|{\boldsymbol {\theta }})\pi ({\boldsymbol {\theta }})}{\pi ({\boldsymbol {y}})}},}$$ the joint posterior distribution of ${\displaystyle {\boldsymbol {x}}}$ and ${\displaystyle {\boldsymbol {\theta }}}$ is given by $${\displaystyle {\begin{aligned}\pi ({\boldsymbol {x}},{\boldsymbol {\theta }}|{\boldsymbol {y}})&\propto \pi ({\boldsymbol {\theta }})\pi ({\boldsymbol {x}}|{\boldsymbol {\theta }})\prod _{i}\pi (y_{i}|\eta _{i},{\boldsymbol {\theta }})\\&\propto \pi ({\boldsymbol {\theta }})\left|{\boldsymbol {Q_{\theta }}}\right|^{1/2}\exp \left(-{\frac {1}{2}}{\boldsymbol {x}}^{T}{\boldsymbol {Q_{\theta }}}{\boldsymbol {x}}+\sum _{i}\log \left\right).\end{aligned}}}$$ Obtaining the exact posterior is generally a very difficult problem. In INLA, the main aim is to approximate the posterior marginals Zdroj:https://en.wikipedia.org?pojem=Integrated_nested_Laplace_approximations
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