Bayes-Newton Methods for Approximate Bayesian Inference with PSD Guarantees

TL;DR

This paper introduces a unified framework called Bayes-Newton that interprets various Bayesian inference algorithms as Newton's method extensions, ensuring positive semi-definite covariance matrices and broad applicability to Gaussian models.

Contribution

It formulates Bayesian inference algorithms as Newton's method extensions, leading to PSD guarantees and unifying different inference schemes under a common optimization perspective.

Findings

Gauss-Newton and quasi-Newton methods are valid within the Bayes-Newton framework.

The new algorithms guarantee PSD covariance matrices, improving stability.

Applications demonstrated on Gaussian processes and state space models.

Abstract

We formulate natural gradient variational inference (VI), expectation propagation (EP), and posterior linearisation (PL) as extensions of Newton's method for optimising the parameters of a Bayesian posterior distribution. This viewpoint explicitly casts inference algorithms under the framework of numerical optimisation. We show that common approximations to Newton's method from the optimisation literature, namely Gauss-Newton and quasi-Newton methods (e.g., the BFGS algorithm), are still valid under this 'Bayes-Newton' framework. This leads to a suite of novel algorithms which are guaranteed to result in positive semi-definite (PSD) covariance matrices, unlike standard VI and EP. Our unifying viewpoint provides new insights into the connections between various inference schemes. All the presented methods apply to any model with a Gaussian prior and non-conjugate likelihood, which we demonstrate with (sparse) Gaussian processes and state space models.