# Laplace Matching for fast Approximate Inference in Latent Gaussian Models

arXiv: https://arxiv.org/abs/2105.03109 · HTML: https://mlsanity.com/paper/2105.03109/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 24 references that are in the map.
- First posted on arXiv 7 May 2021.

## Builds on (most cited first, 4 of 4)

- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Dirichlet-based Gaussian Processes for Large-scale Calibrated Classification](https://mlsanity.com/paper/1805.10915/) (2018)
- [Fast Predictive Uncertainty for Classification with Bayesian Deep Networks](https://mlsanity.com/paper/2003.01227/) (2020)
- [Latent Gaussian Count Time Series](https://mlsanity.com/paper/1811.00203/) (2018)

## Built on by (most cited first, 4 of 4)

- [Flexible inference in heterogeneous and attributed multilayer networks](https://mlsanity.com/paper/2405.20918/) (2024)
- [Geometric Gaussian Approximations of Probability Distributions](https://mlsanity.com/paper/2507.00616/) (2025)
- [Likelihood approximations via Gaussian approximate inference](https://mlsanity.com/paper/2410.20754/) (2024)
- [How Low Can You Go: Fine-Tuning LLMs with Bayesian Optimization (LlMBO)](https://mlsanity.com/paper/2610.18504/) (2026)

## Abstract (arXiv)

Bayesian inference on non-Gaussian data is often non-analytic and requires computationally expensive approximations such as sampling or variational inference. We propose an approximate inference framework primarily designed to be computationally cheap while still achieving high approximation quality. The concept, which we call Laplace Matching, involves closed-form, approximate, bi-directional transformations between the parameter spaces of exponential families. These are constructed from Laplace approximations under custom-designed basis transformations. The mappings can then be leveraged to effectively turn a latent Gaussian distribution into an approximate conjugate prior to a rich class of observable variables. This allows us to train latent Gaussian models such as Gaussian Processes on non-Gaussian data at nearly no additional cost. The method can be thought of as a pre-processing step which can be implemented in <5 lines of code and runs in less than a second. Furthermore, Laplace Matching yields a simple way to group similar data points together, e.g. to produce inducing points for GPs. We empirically evaluate the method with experiments for four different exponential distributions, namely the Beta, Gamma, Dirichlet and inverse Wishart, showing approximation quality comparable to state-of-the-art approximate inference techniques at a drastic reduction in computational cost.