Importance Sampling with the Integrated Nested Laplace Approximation

TL;DR

This paper introduces importance sampling methods combined with INLA for Bayesian inference, demonstrating improved accuracy and robustness in various models, especially with the adaptive AMIS-INLA approach.

Contribution

It proposes combining importance sampling with INLA and extends it with an adaptive algorithm, enhancing inference robustness and performance in complex Bayesian models.

Findings

AMIS-INLA outperforms other methods in accuracy and robustness.

IS-INLA offers faster inference with good proposals.

Validated on multiple Bayesian models with simulated and real data.

Abstract

The Integrated Nested Laplace Approximation (INLA) is a deterministic approach to Bayesian inference on latent Gaussian models (LGMs) and focuses on fast and accurate approximation of posterior marginals for the parameters in the models. Recently, methods have been developed to extend this class of models to those that can be expressed as conditional LGMs by fixing some of the parameters in the models to descriptive values. These methods differ in the manner descriptive values are chosen. This paper proposes to combine importance sampling with INLA (IS-INLA), and extends this approach with the more robust adaptive multiple importance sampling algorithm combined with INLA (AMIS-INLA). This paper gives a comparison between these approaches and existing methods on a series of applications with simulated and observed datasets and evaluates their performance based on accuracy, efficiency, and robustness. The approaches are validated by exact posteriors in a simple bivariate linear model; then, they are applied to a Bayesian lasso model, a Bayesian imputation of missing covariate values, and lastly, in parametric Bayesian quantile regression. The applications show that the AMIS-INLA approach, in general, outperforms the other methods, but the IS-INLA algorithm could be considered for faster inference when good proposals are available.