Data Analysis Recipes: Products of multivariate Gaussians in Bayesian inferences

TL;DR

This paper derives closed-form expressions for the product of Gaussian likelihoods and priors in linear Bayesian models, enabling faster and more precise inference in physics and astronomy applications.

Contribution

It provides new closed-form formulas for Gaussian products in linear Bayesian models, simplifying inference procedures.

Findings

Derived explicit formulas for Gaussian products in Bayesian inference.

Demonstrated improved inference speed and accuracy with these formulas.

Connected methods to practical physics and astronomy problems.

Abstract

A product of two Gaussians (or normal distributions) is another Gaussian. That's a valuable and useful fact! Here we use it to derive a refactoring of a common product of multivariate Gaussians: The product of a Gaussian likelihood times a Gaussian prior, where some or all of those parameters enter the likelihood only in the mean and only linearly. That is, a linear, Gaussian, Bayesian model. This product of a likelihood times a prior pdf can be refactored into a product of a marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where (in this case) both of these are also Gaussian. The means and variance tensors of the refactored Gaussians are straightforward to obtain as closed-form expressions; here we deliver these expressions, with discussion. The closed-form expressions can be used to speed up and improve the precision of inferences that contain linear parameters with Gaussian priors. We connect these methods to inferences that arise frequently in physics and astronomy. If all you want is the answer, the question is posed and answered at the beginning of Section 3. We show two toy examples, in the form of worked exercises, in Section 4. The solutions, discussion, and exercises in this Note are aimed at someone who is already familiar with the basic ideas of Bayesian inference and probability.