GMRFs in JuliaFast and Flexible Latent Gaussian Modelling

What are Gaussian Markov Random Fields? ​

A Gaussian Markov Random Field (GMRF) is a Gaussian distribution whose precision matrix — the inverse of the covariance — is sparse.

Sparsity in the precision is a modelling statement. A zero in entry $(i,j)$ of the precision matrix says precisely that $x_i$ and $x_j$ are conditionally independent given all the other variables. Most quantities we model in space and time behave that way: the temperature here depends on the temperature next door, and only indirectly on the temperature a hundred kilometres away. Encoding that structure in the precision matrix is what makes the field Markov.

This brings nice computational benefits. The covariance matrix of such a distribution is usually completely dense, so working with it directly costs $O(n^3)$ and becomes hopeless as $n$ grows. The precision matrix is sparse, and sparse Cholesky factorization takes advantage of that — which is how GMRFs reach problem sizes where a naive Gaussian process cannot keep up.

The difficulty has always been that the interesting priors are awkward to write down in precision form. One answer for spatial problems is the SPDE approach: state the model as a stochastic partial differential equation, discretize it with finite elements, and obtain a GMRF that approximates it. Getting started walks through both a hand-written precision matrix and an SPDE-derived one.

Quick Start ​

Install the package:

using Pkg
Pkg.add("GaussianMarkovRandomFields")
using GaussianMarkovRandomFields

Related packages ​

Latte.jl is a probabilistic programming language for latent Gaussian models, built by the same author on top of this package. It provides INLA, TMB-style Laplace approximations and HMC-Laplace behind a concise model syntax. If you want a complete inference workflow rather than the components to build one, start there; this package is what it uses underneath.

Getting Help ​

Questions, bug reports and feature requests all belong in the issue tracker. Please open an issue rather than emailing the maintainer, so that answers stay searchable for everyone.

The contribution guidelines describe what to include in an issue, how to contribute code, and what to expect regarding response times and project scope.

Citing ​

If you use GaussianMarkovRandomFields.jl in your research, please cite it via its Zenodo archive. That DOI always resolves to the latest release, and the Zenodo page offers BibTeX and other export formats as well as per-version DOIs.

@software{weiland_gmrf_jl,
  author    = {Weiland, Tim},
  title     = {{GaussianMarkovRandomFields.jl}},
  publisher = {Zenodo},
  year      = {2025},
  doi       = {10.5281/zenodo.18088214},
  url       = {https://doi.org/10.5281/zenodo.18088214}
}