Quantum Probability for Statisticians: Some New Ideas

It is argued from several points of view that quantum probabilities might play a role in statistical settings. New approaches toward quantum foundations have postulates that appear to be equally valid in macroscopic settings. One such approach is described here in detail, while two others are briefly sketched. In particular, arguments behind the Born rule, which gives the basis for quantum probabilities, are given. A list of ideas for possible statistical applications of quantum probabilities is provided and discussed. A particular area is machine learning, where there exists substantial literature on links to quantum probability. Here, an idea about model reduction is sketched and is motivated from a quantum probability model. Quantum models can play a role in model reduction, where the partial least squares regression model is a special case. It is shown that for certain experiments, a Bayesian prior given by a quantum probability can be motivated. Quantum decision theory is an emerging discipline that can be motivated by this author’s theory of quantum foundations.