Bayesianism, Conditional Probability and Laplace Law of Succession in Quantum Mechanics

TL;DR

This paper explores the relationship between Bayesian probability and quantum probability, demonstrating that quantum probability can be coherently interpreted as rational belief, similar to classical probability, with a modified Laplace law for small sample sizes.

Contribution

It shows that Bayesian probability principles apply to quantum mechanics, with a slight modification to the Laplace law of succession for small observations.

Findings

Quantum probability aligns with Bayesian rational belief.

Modified Laplace law applies in quantum context for few observations.

Bayesian and frequency interpretations are consistent in quantum mechanics.

Abstract

We present a comparative study between classical probability and quantum probability from the Bayesian viewpoint, where probability is construed as our rational degree of belief on whether a given statement is true. From this viewpoint, including conditional probability, three issues are discussed: i) Given a measure of the rational degree of belief, does it satisfy the axioms of the probability? ii) Given the probability satisfying these axioms, is it seen as the measure of the rational degree of belief? iii) Can the measure of the rational degree of belief be evaluated in terms of the relative frequency of events occurring? Here we show that as with the classical probability, all these issues can be resolved affirmatively in the quantum probability, provided that the relation to the relative frequency is slightly modified from the Laplace law of succession in case of a small number of observations. This implies that the relation between the Bayesian probability and the relative frequency in quantum mechanics is the same as that in the classical probability theory, including conditional probability.