A quantum framework for likelihood ratios

TL;DR

This paper introduces a quantum information-theoretic framework for calculating likelihood ratios, addressing limitations of classical methods and unifying classifiers under a quantum perspective.

Contribution

It develops a novel quantum-based formula for likelihood ratios, demonstrating that classical classifiers are special cases within this quantum framework.

Findings

Likelihood ratio is a fundamental property of statistical systems.

Naive Bayes' classifier is a special case of the quantum approach.

Quantum methods can overcome classical axiomatic limitations.

Abstract

The ability to calculate precise likelihood ratios is fundamental to many STEM areas, such as decision-making theory, biomedical science, and engineering. However, there is no assumption-free statistical methodology to achieve this. For instance, in the absence of data relating to covariate overlap, the widely used Bayes' theorem either defaults to the marginal probability driven "naive Bayes' classifier", or requires the use of compensatory expectation-maximization techniques. Equally, the use of alternative statistical approaches, such as multivariate logistic regression, may be confounded by other axiomatic conditions, e.g., low levels of co-linearity. This article takes an information-theoretic approach in developing a new statistical formula for the calculation of likelihood ratios based on the principles of quantum entanglement. In doing so, it is argued that this quantum approach demonstrates: that the likelihood ratio is a real quality of statistical systems; that the naive Bayes' classifier is a special case of a more general quantum mechanical expression; and that only a quantum mechanical approach can overcome the axiomatic limitations of classical statistics.