Consistent reasoning about a continuum of hypotheses on the basis of finite evidence

TL;DR

This paper explores how classical Bayesian reasoning must be adapted for hypotheses forming a continuum when evidence is finite, revealing that quantum theory's structure naturally emerges as the consistent framework.

Contribution

It demonstrates that the only consistent reasoning framework for a continuum of hypotheses with finite evidence is isomorphic to quantum theory in complex Hilbert space.

Findings

Quantum theory's symmetry group is derived from consistency requirements.

Classical probability theory is insufficient for continuous hypothesis spaces with finite evidence.

The modified reasoning framework aligns with the mathematical structure of quantum mechanics.

Abstract

In the modern Bayesian view classical probability theory is simply an extension of conventional logic, i.e., a quantitative tool that allows for consistent reasoning in the presence of uncertainty. Classical theory presupposes, however, that--at least in principle--the amount of evidence that an experimenter can accumulate always matches the size of the hypothesis space. I investigate how the framework for consistent reasoning must be modified in non-classical situations where hypotheses form a continuum, yet the maximum evidence accessible through experiment is not allowed to exceed some finite upper bound. Invoking basic consistency requirements pertaining to the preparation and composition of systems, as well as to the continuity of probabilities, I show that the modified theory must have an internal symmetry isomorphic to the unitary group. It thus appears that the only consistent algorithm for plausible reasoning about a continuum of hypotheses on the basis of finite evidence is furnished by quantum theory in complex Hilbert space.