[Paper Review] inlabru: software for fitting latent Gaussian models with non-linear predictors

Finn Lindgren, Fabian E. Bachl+4 more|arXiv (Cornell University)|Jun 30, 2024

Simulation Techniques and ApplicationsDecision Sciences3 citations

TL;DR

This paper introduces inlabru, an R package that extends INLA to fit latent Gaussian models with non-linear predictors by iteratively linearizing the predictor using first-order Taylor expansions and fitting each linearized configuration via INLA. The method enables efficient Bayesian inference for complex models—such as spatial point processes, non-linear detection functions, and self-exciting point processes—that are otherwise infeasible with standard INLA.

ABSTRACT

The integrated nested Laplace approximation (INLA) method has become a popular approach for computationally efficient approximate Bayesian computation. In particular, by leveraging sparsity in random effect precision matrices, INLA is commonly used in spatial and spatio-temporal applications. However, the speed of INLA comes at the cost of restricting the user to the family of latent Gaussian models and the likelihoods currently implemented in {INLA}, the main software implementation of the INLA methodology. {inlabru} is a software package that extends the types of models that can be fitted using INLA by allowing the latent predictor to be non-linear in its parameters, moving beyond the additive linear predictor framework to allow more complex functional relationships. For inference it uses an approximate iterative method based on the first-order Taylor expansion of the non-linear predictor, fitting the model using INLA for each linearised model configuration. {inlabru} automates much of the workflow required to fit models using {R-INLA}, simplifying the process for users to specify, fit and predict from models. There is additional support for fitting joint likelihood models by building each likelihood individually. {inlabru} also supports the direct use of spatial data structures, such as those implemented in the {sf} and {terra} packages. In this paper we outline the statistical theory, model structure and basic syntax required for users to understand and develop their own models using {inlabru}. We evaluate the approximate inference method using a Bayesian method checking approach. We provide three examples modelling simulated spatial data that demonstrate the benefits of the additional flexibility provided by {inlabru}.

Motivation & Objective

• To extend the class of models amenable to fast Bayesian inference using INLA by allowing non-linear predictors.
• To address the limitation of standard INLA, which restricts predictors to be linear in latent parameters.
• To automate model specification, fitting, and prediction workflows in R, reducing user burden.
• To support advanced modeling features such as joint likelihoods, spatial data structures (sf, terra), and integration with add-on packages.
• To enable application of INLA’s computational efficiency to complex ecological, epidemiological, and geospatial models with non-linear relationships.

Proposed method

• Uses first-order Taylor expansion to linearize non-linear predictors around current parameter estimates.
• Applies INLA iteratively to fit successive linearized approximations of the model.
• Automates the 'stack building' process in INLA through high-level R syntax, abstracting low-level model configuration.
• Supports direct use of spatial data objects (sf, terra, sp, raster) in model specification.
• Provides modular hooks for integrating with add-on packages (e.g., ETAS.inlabru, dirinla, rSPDE) to extend model types.
• Offers predict() and generate() methods for streamlined post-estimation inference and simulation.

Experimental results

Research questions

• RQ1 Can iterative linearization via Taylor expansion enable efficient Bayesian inference for latent Gaussian models with non-linear predictors?
• RQ2 How does the performance of inlabru compare to exact MCMC or other approximate methods in terms of accuracy and speed?
• RQ3 To what extent can inlabru support complex, real-world models such as self-exciting point processes or non-linear detection functions?
• RQ4 How well does the iterative method converge to the true posterior distribution in practice?
• RQ5 What is the practical impact of inlabru on model development in fields like ecology, public health, and seismology?

Key findings

• The iterative linearization method in inlabru achieves good convergence to the true posterior distribution in under 30 adaptations of the proposal distribution.
• inlabru enables fitting of models with non-linear predictors—such as half-normal detection functions and self-exciting Hawkes processes—that are outside the scope of standard INLA.
• The package successfully supports joint likelihood models by allowing individual likelihoods to be built and combined, enhancing flexibility.
• inlabru integrates seamlessly with spatial data formats (sf, terra) and supports advanced modeling features like random fields on graphs and fractional SPDEs.
• The method has been successfully applied in published studies involving animal telemetry, seismic activity, and dolphin abundance modeling.
• inlabru maintains compatibility with new INLA features, including inla.mdata and inla.surv, ensuring ongoing support for evolving INLA capabilities.