Bayesian statistical learning using density operators

TL;DR

This paper introduces a quantum mechanics-inspired reformulation of Bayesian learning using density operators, enabling new coordinate system formulations and kernel-based learning of quantum states.

Contribution

It presents a novel approach that uses density operators in Bayesian learning, allowing for wave function decomposition and kernel methods to preserve quantum probabilistic structure.

Findings

Density operators enable alternative coordinate system formulations.

Kernel tricks can be applied to learn projections of density operators.

Wave function decomposition preserves the nature of probability operators.

Abstract

This short study reformulates the statistical Bayesian learning problem using a quantum mechanics framework. Density operators representing ensembles of pure states of sample wave functions are used in place probability densities. We show that such representation allows to formulate the statistical Bayesian learning problem in different coordinate systems on the sample space. We further show that such representation allows to learn projections of density operators using a kernel trick. In particular, the study highlights that decomposing wave functions rather than probability densities, as it is done in kernel embedding, allows to preserve the nature of probability operators. Results are illustrated with a simple example using discrete orthogonal wavelet transform of density operators.