Bayes-Newton is a library for approximate inference in Gaussian processes (GPs) in JAX (with objax), built and maintained by Will Wilkinson.

Bayes-Newton provides a unifying view of approximate Bayesian inference, and allows for the combination of many models (e.g. GPs, sparse GPs, Markov GPs, sparse Markov GPs) with the inference method of your choice (VI, EP, Laplace, Linearisation). For a full list of the methods implemented scroll down to the bottom of this page.

The methodology is outlined in the following article:

• W.J. Wilkinson, S. Särkkä, and A. Solin: Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees. JMLR volume 24 (2023).

Latest (stable) release from PyPI

pip install bayesnewton

For development, you might want to use the latest source from GitHub: In a check-out of the develop branch of the BayesNewton GitHub repository, run

pip install -e .

For running the demos or experiments in this repository or building on top of it, you can follow these steps for creating a virtual environment and activating it:

python3 -m venv venv
source venv/bin/activate

Installing all required dependencies for the examples:

python -m pip install -r requirements.txt
python -m pip install -e .

Running the tests requires additionally a specific version of GPflow to test against:

python -m pip install pytest
python -m pip install tensorflow==2.10 tensorflow-probability==0.18.0 gpflow==2.5.2

Run tests

cd tests; pytest

Given some inputs x and some data y, you can construct a Bayes-Newton model as follows,

kern = bayesnewton.kernels.Matern52()
lik = bayesnewton.likelihoods.Gaussian()
model = bayesnewton.models.MarkovVariationalGP(kernel=kern, likelihood=lik, X=x, Y=y)

The training loop (inference and hyperparameter learning) is then set up using objax's built in functionality:

lr_adam = 0.1
lr_newton = 1
opt_hypers = objax.optimizer.Adam(model.vars())
energy = objax.GradValues(model.energy, model.vars())

@objax.Function.with_vars(model.vars() + opt_hypers.vars())
def train_op():
    model.inference(lr=lr_newton, **inf_args)  # perform inference and update variational params
    dE, E = energy(**inf_args)  # compute energy and its gradients w.r.t. hypers
    opt_hypers(lr_adam, dE)  # update the hyperparameters
    return E

As we are using JAX, we can JIT compile the training loop:

train_op = objax.Jit(train_op)

and then run the training loop,

iters = 20
for i in range(1, iters + 1):
    loss = train_op()

Full demos are available here.

@article{wilkinson2023bayes,
  title={{B}ayes--{N}ewton Methods for Approximate {B}ayesian Inference with {PSD} Guarantees},
  author={Wilkinson, William J and S{\"a}rkk{\"a}, Simo and Solin, Arno},
  journal={Journal of Machine Learning Research},
  volume={24},
  number={83},
  pages={1--50},
  year={2023}
}

For a full list of the all the models available see the model class list.

• Variationl GP (Opper, Archambeau: The Variational Gaussian Approximation Revisited, Neural Computation 2009; Khan, Lin: Conugate-Computation Variational Inference - Converting Inference in Non-Conjugate Models in to Inference in Conjugate Models, AISTATS 2017)
• Sparse Variational GP (Hensman, Matthews, Ghahramani: Scalable Variational Gaussian Process Classification, AISTATS 2015; Adam, Chang, Khan, Solin: Dual Parameterization of Sparse Variational Gaussian Processes, NeurIPS 2021)
• Markov Variational GP (Chang, Wilkinson, Khan, Solin: Fast Variational Learning in State Space Gaussian Process Models, MLSP 2020)
• Sparse Markov Variational GP (Adam, Eleftheriadis, Durrande, Artemev, Hensman: Doubly Sparse Variational Gaussian Processes, AISTATS 2020; Wilkinson, Solin, Adam: Sparse Algorithms for Markovian Gaussian Processes, AISTATS 2021)
• Spatio-Temporal Variational GP (Hamelijnck, Wilkinson, Loppi, Solin, Damoulas: Spatio-Temporal Variational Gaussian Processes, NeurIPS 2021)

• Expectation Propagation GP (Minka: A Family of Algorithms for Approximate Bayesian Inference, Ph. D thesis 2000)
• (Stochastic) Sparse Expectation Propagation GP (Csato, Opper: Sparse on-line Gaussian processes, Neural Computation 2002; Bui, Yan, Turner: A Unifying Framework for Gaussian Process Pseudo Point Approximations Using Power Expectation Propagation, JMLR 2017)
• Markov Expectation Propagation GP (Wilkinson, Chang, Riis Andersen, Solin: State Space Expectation Propagation, ICML 2020)
• Sparse Markov Expectation Propagation GP (Wilkinson, Solin, Adam: Sparse Algorithms for Markovian Gaussian Processes, AISTATS 2021)

• Laplace GP (Rasmussen, Williams: Gaussian Processes for Machine Learning, 2006)
• Sparse Laplace GP (Wilkinson, Särkkä, Solin: Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees)
• Markov Laplace GP (Wilkinson, Särkkä, Solin: Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees)
• Sparse Markov Laplace GP

• Posterior Linearisation GP (García-Fernández, Tronarp, Sarkka: Gaussian Process Classification Using Posterior Linearization, IEEE Signal Processing 2019; Steinberg, Bonilla: Extended and Unscented Gaussian Processes, NeurIPS 2014)
• Sparse Posterior Linearisation GP
• Markov Posterior Linearisation GP (García-Fernández, Svensson, Sarkka: Iterated Posterior Linearization Smoother, IEEE Automatic Control 2016; Wilkinson, Chang, Riis Andersen, Solin: State Space Expectation Propagation, ICML 2020)
• Sparse Markov Posterior Linearisation GP (Wilkinson, Solin, Adam: Sparse Algorithms for Markovian Gaussian Processes, AISTATS 2021)
• Taylor Expansion / Analytical Linearisaiton GP (Steinberg, Bonilla: Extended and Unscented Gaussian Processes, NeurIPS 2014)
• Markov Taylor GP / Extended Kalman Smoother (Bell: The Iterated Kalman Smoother as a Gauss-Newton method, SIAM Journal on Optimization 1994)
• Sparse Taylor GP
• Sparse Markov Taylor GP / Sparse Extended Kalman Smoother (Wilkinson, Solin, Adam: Sparse Algorithms for Markovian Gaussian Processes, AISTATS 2021)

(Wilkinson, Särkkä, Solin: Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees)

• Gauss-Newton
• Variational Gauss-Newton
• PEP Gauss-Newton
• 2nd-order PL Gauss-Newton

(Wilkinson, Särkkä, Solin: Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees)

• Quasi-Newton
• Variational Quasi-Newton
• PEP Quasi-Newton
• PL Quasi-Newton

• VI Riemann Grad (Lin, Schmidt, Khan: Handling the Positive-Definite Constraint in the Bayesian Learning Rule, ICML 2020)
• Newton/Laplace Riemann Grad (Lin, Schmidt, Khan: Handling the Positive-Definite Constraint in the Bayesian Learning Rule, ICML 2020)
• PEP Riemann Grad (Wilkinson, Särkkä, Solin: Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees)

• Infinite Horizon GP (Solin, Hensman, Turner: Infinite-Horizon Gaussian Processes, NeurIPS 2018)
• Parallel Markov GP (with VI, EP, PL, ...) (Särkkä, García-Fernández: Temporal parallelization of Bayesian smoothers; Corenflos, Zhao, Särkkä: Gaussian Process Regression in Logarithmic Time; Hamelijnck, Wilkinson, Loppi, Solin, Damoulas: Spatio-Temporal Variational Gaussian Processes, NeurIPS 2021)
• 2nd-order Posterior Linearisation GP (sparse, Markov, ...) (Wilkinson, Särkkä, Solin: Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees)

This software is provided under the Apache License 2.0. See the accompanying LICENSE file for details.