Approximate Bayesian inference based on INLA algorithm

The integrated nested Laplace approximation (INLA) algorithm provides a computationally efficient approach for approximate Bayesian inference, overcoming the limitations of traditional Markov chain Monte Carlo (MCMC) methods. This paper reviews INLA algorithm and provides a systematic review of six key books that explore the theoretical foundations, practical implementations, and diverse applications of INLA. These six books cover spatial and spatio-temporal modelling, general Bayesian inference, SPDE-based spatial analysis, geospatial health data, regression modelling, and dynamic time series. In addition, these books highlight the versatility of INLA method in handling complex models while maintaining high computational efficiency. This paper begins with an introduction to the INLA method and algorithm, followed by a systematic review of six key publications in the field.