# Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models

arXiv: https://arxiv.org/abs/2605.20345 · HTML: https://mlsanity.com/paper/2605.20345/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 45 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 19 of 19)

- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Importance Weighting and Variational Inference](https://mlsanity.com/paper/1808.09034/) (2018)
- [Yes, but Did It Work?: Evaluating Variational Inference](https://mlsanity.com/paper/1802.02538/) (2018)
- [Hamiltonian Monte Carlo for Hierarchical Models](https://mlsanity.com/paper/1312.0906/) (2013)
- [BlackJAX: Composable Bayesian inference in JAX](https://mlsanity.com/paper/2402.10797/) (2024)
- [Automatic Reparameterisation of Probabilistic Programs](https://mlsanity.com/paper/1906.03028/) (2019)
- [Black Box Variational Inference with a Deterministic Objective: Faster, More Accurate, and Even More Black Box](https://mlsanity.com/paper/2304.05527/) (2023)
- [Bayesian Computing with INLA: A Review](https://mlsanity.com/paper/1604.00860/) (2016)
- [Divide and Couple: Using Monte Carlo Variational Objectives for Posterior Approximation](https://mlsanity.com/paper/1906.10115/) (2019)
- [The Kernel Interaction Trick: Fast Bayesian Discovery of Pairwise Interactions in High Dimensions](https://mlsanity.com/paper/1905.06501/) (2019)
- [Pseudo-Marginal Hamiltonian Monte Carlo](https://mlsanity.com/paper/1607.02516/) (2016)
- [Model-Informed Flows for Bayesian Inference](https://mlsanity.com/paper/2505.24243/) (2025)
- [Generalized Guarantees for Variational Inference in the Presence of Even and Elliptical Symmetry](https://mlsanity.com/paper/2511.01064/) (2025)
- [Improving the INLA approach for approximate Bayesian inference for latent Gaussian models](https://mlsanity.com/paper/1503.07307/) (2015)
- [Importance Sampling with the Integrated Nested Laplace Approximation](https://mlsanity.com/paper/2103.02721/) (2021)
- [An Extended Simplified Laplace strategy for Approximate Bayesian inference of Latent Gaussian Models using R-INLA](https://mlsanity.com/paper/2203.14304/) (2022)
- [Hamiltonian Monte Carlo Inference of Marginalized Linear Mixed-Effects Models](https://mlsanity.com/paper/2410.24079/) (2024)

## Abstract (arXiv)

Latent Gaussian models (LGMs) are a popular class of Bayesian hierarchical models that include Gaussian processes, as well as certain spatial models and mixed-effect models. Efficient Bayesian inference of LGMs often requires marginalizing out the latent variables. For LGMs with a non-Gaussian likelihood, exact marginalization is not possible and a popular approach is to do approximate marginalization with an integrated Laplace approximation (ILA). Using ILA produces an approximate posterior which, in some settings, can differ significantly from the correct posterior, which impacts downstream applications. We propose an importance sampling scheme to correct the error introduced by ILA. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. We implement our methods in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters. For the latter, we specifically consider the use of Hamiltonian Monte Carlo. We demonstrate the benefits of reduced error in various applied models.