标题： 为什么是概率的平方根？ 显示英文标题

标题： Why Square Roots of Probabilities?

摘要： 概率的平方根出现在多个上下文中，这表明它们在某种意义上比概率本身更根本。概率的平方根出现在费雪-拉奥度量和海林格-巴塔查里亚距离的表达式中。 它们还通过玻恩规则在量子力学中发挥作用，在量子力学中，概率是通过量子振幅的模平方来确定的。 为什么会这样？为什么这些平方根不会出现在概率论的各种表述中？ 在这篇简短且未完成的探索中，我考虑用一个由一组分量定义的向量来量化逻辑陈述，每个分量量化定义假设空间的原子陈述之一。 我证明了条件概率（双赋值），如$P(x|y)$，可以写成两个向量的点积，这两个向量分别量化逻辑陈述$x$和$y$，并且每个都相对于量化条件的向量$y$进行归一化。 向量的分量与概率的平方根成正比。 因此，这种表述被证明与应用于互斥原子陈述集合的正交性概念是一致的，使得总和规则表示为概率平方根的平方和。 显示英文摘要

摘要： Square roots of probabilities appear in several contexts, which suggests that they are somehow more fundamental than probabilities. Square roots of probabilities appear in expressions of the Fisher-Rao Metric and the Hellinger-Bhattacharyya distance. They also come into play in Quantum Mechanics via the Born rule where probabilities are found by taking the squared modulus of the quantum amplitude. Why should this be the case and why do these square roots not arise in the various formulations of probability theory? In this short, inconclusive exploration, I consider quantifying a logical statement with a vector defined by a set of components each quantifying one of the atomic statements defining the hypothesis space. I show that conditional probabilities (bi-valuations), such as $P(x|y)$, can be written as the dot product of the two vectors quantifying the logical statements $x$ and $y$ each normalized with respect to the vector quantifying the conditional $y$. The components of the vectors are proportional to the square root of the probability. As a result, this formulation is shown to be consistent with a concept of orthogonality applied to the set of mutually exclusive atomic statements such that the sum rule is represented as the sum of the squares of the square roots of probability.