### Abstract

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 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.

Original language | English |
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Pages (from-to) | 395-421 |

Number of pages | 27 |

Journal | Annual Review of Statistics and Its Application |

Volume | 4 |

Early online date | 23 Dec 2016 |

DOIs | |

Publication status | Published - 7 Mar 2017 |

### Keywords

- Approximate Bayesian inference
- Gaussian Markov random fields
- Laplace approximations
- Latent Gaussian models
- Numerical integration
- Sparse matrices

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## Cite this

*Annual Review of Statistics and Its Application*,

*4*, 395-421. https://doi.org/10.1146/annurev-statistics-060116-054045