⚡ Fast, flexible and user-centered Julia package for Bayesian inference with sparse Gaussians
Gaussian Markov Random Fields (GMRFs) are Gaussian distributions with sparse precision (inverse covariance) matrices. GaussianMarkovRandomFields.jl provides utilities for working with GMRFs in Julia. The goal is to enable flexible and efficient Bayesian inference from GMRFs, powered by sparse linear algebra.
In particular, we support the creation of GMRFs through finite element method discretizations of stochastic partial differential equations (SPDEs). This unlocks efficient GMRF-based approximations to commonly used Gaussian process priors. Furthermore, the expressive power of SPDEs allows for flexible, problem-tailored priors.
GaussianMarkovRandomFields.jl is a registered Julia package.
To install it, launch the Julia REPL and type ] add GaussianMarkovRandomFields.
Let's construct a GMRF approximation to a Matérn process from observation points:
using GaussianMarkovRandomFields
# Define observation points
points = [0.1 0.0; -0.3 0.55; 0.2 0.8; -0.1 -0.2] # N×2 matrix
# Create Matérn latent model (automatically generates mesh and discretization)
model = MaternModel(points; smoothness = 1)
x = model(τ = 1.0, range = 0.3) # Construct GMRF with specified parametersx is a Gaussian distribution, and we can compute all the things Gaussians are
known for.
# Get interesting quantities
μ = mean(x)
σ_marginal = std(x)
samp = rand(x) # Sample
Q = precision_map(x) # Sparse precision matrix
# Form posterior under point observations
using Distributions: Normal
obs_model = PointEvaluationObsModel(model, Normal)
y = [0.83, 0.12, 0.45, -0.21]
obs_likelihood = obs_model(y; σ = 0.1)
x_cond = gaussian_approximation(x, obs_likelihood) # Posterior GMRF!Make sure to check the documentation for further examples!
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.
See Getting Help for what to include, and Support and Governance for what to expect regarding response times and project scope.
If you use GaussianMarkovRandomFields.jl in your research, please cite it via its Zenodo archive. The DOI below always resolves to the latest release; the Zenodo page additionally lets you cite a specific version and offers BibTeX and other export formats.
Tim Weiland. GaussianMarkovRandomFields.jl. Zenodo. https://doi.org/10.5281/zenodo.18088214
@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}
}GitHub's Cite this repository button exposes the same metadata, which lives in
CITATION.cff.
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