Bayesian statistical learning using density operators

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.