Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models

2026-05-21 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, long

Summary

This research introduces corrected Integrated Laplace Approximation (ILA) methods for Bayesian inference in Latent Gaussian Models (LGMs), which are widely used in Gaussian processes and mixed-effect models. While ILA efficiently marginalizes latent variables, it can introduce significant posterior approximation errors, especially with non-Gaussian likelihoods. The authors propose an importance sampling scheme to correct this error, ensuring convergence to the true posterior as the number of samples increases. This scheme is realized through three techniques: pseudo-marginalization (PM-ADLA), quasi-Monte Carlo (QMC-ADLA), and randomized quasi-Monte Carlo (RQMC-ADLA). These methods are implemented within an automatic differentiation framework, specifically JAX, to support gradient-based algorithms like Hamiltonian Monte Carlo (HMC) for high-dimensional hyperparameters, demonstrating reduced error and computational benefits across various applied models.

Key takeaway

For AI Scientists and Machine Learning Engineers working with Latent Gaussian Models, especially those encountering posterior approximation errors from Integrated Laplace Approximation, you should consider adopting these importance sampling-based correction methods. Implementing PM-ADLA, QMC-ADLA, or RQMC-ADLA can provide asymptotically correct posteriors while retaining the computational advantages of marginalization, significantly mitigating issues like divergent HMC transitions and improving inference accuracy in complex models.

Key insights

Importance sampling corrects Integrated Laplace Approximation errors for asymptotically exact Bayesian posteriors in LGMs.

Principles

• ILA can be reformulated as a single-sample importance sampling estimator.
• Increasing importance samples reduces estimation error, converging to zero.
• Quasi-Monte Carlo (QMC) reduces variance more efficiently than raw importance sampling.

Method

The method uses importance sampling to correct ILA error, implemented via pseudo-marginalization, quasi-Monte Carlo, or randomized QMC within an automatic differentiation framework (JAX) to enable gradient-based HMC.

In practice

• Apply PM-ADLA to correct posterior error in LGMs with high-dimensional hyperparameters.
• Utilize QMC-ADLA for more efficient variance reduction in importance sampling.
• Implement methods in JAX for efficient gradient computation with HMC.

Topics

• Bayesian Inference
• Latent Gaussian Models
• Laplace Approximation
• Importance Sampling
• Hamiltonian Monte Carlo
• Automatic Differentiation
• Quasi-Monte Carlo

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.