  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.
