If there are two dependent positive real variables ${\mathfrak{x}}_1$ and ${\mathfrak{x}}_2$, and only ${\mathfrak{x}}_1$ is known, what is the probability that ${\mathfrak{x}}_2$ is larger versus smaller than ${\mathfrak{x}}_1$? There is no uniquely correct answer according to “frequentist” and “subjective Bayesian” definitions of probability. Here we derive the answer given the “objective Bayesian” definition developed by Jeffreys, Cox, and Jaynes. We declare the standard distance metric in one dimension, $d(A,B)\equiv |A-B|$, and the uniform prior distribution, as axioms. If neither variable is known, $P(x_2<x_1)=P(x_2>x_1)$. This appears obvious, since the state spaces $x_2<x_1$ and $x_2>x_1$ have equal size. However, if ${\mathfrak{x}}_1$ is known and ${\mathfrak{x}}_2$ unknown, there are infinitely more numbers in the space $x_2>{\mathfrak{x}}_1$ than $x_2<{\mathfrak{x}}_1$. Despite this asymmetry, we prove $P(x_2<{\mathfrak{x}}_1\mid {\mathfrak{x}}_1)=P(x_2>{\mathfrak{x}}_1\mid {\mathfrak{x}}_1)$, so that ${\mathfrak{x}}_1$ is the median of $p\left(x_2\right|{\mathfrak{x}}_1)$, and ${\mathfrak{x}}_1$ is statistically independent of ratio ${\mathfrak{x}}_2/{\mathfrak{x}}_1$. We present three proofs that apply to all members of a set of distributions. Each member is distinguished by the form of dependence between variables implicit within a statistical model (gamma, Gaussian, etc.), but all exhibit two symmetries in the joint distribution $p(x_1,x_2)$ that are required in the absence of prior information: exchangeability of variables, and non-informative priors over the marginal distributions $p\left(x_1\right)$ and $p\left(x_2\right)$. We relate our conclusion to physical models of prediction and intelligence, where the known ’sample’ could be the present internal energy within a sensor, and the unknown the energy in its external sensory cause or future motor effect.

中文翻译：

---

未知的正实变量大于已知的客观贝叶斯概率为1/2

如果有两个从属正实变量 ${\mathfrak{X}}_{\mathrm{1个}}$ 和 ${\mathfrak{X}}_{\mathrm{2个}}$，并且只有 ${\mathfrak{X}}_{\mathrm{1个}}$ 是已知的，概率是多少 ${\mathfrak{X}}_{\mathrm{2个}}$ 大于小于 ${\mathfrak{X}}_{\mathrm{1个}}$？根据概率的“频率”和“主观贝叶斯”定义，没有唯一正确的答案。在这里，我们根据杰弗里斯（Jeffreys），考克斯（Cox）和贾恩斯（Jaynes）提出的“客观贝叶斯”定义得出了答案。我们在一维中声明标准距离度量，$d（\mathrm{一个}，乙）\equiv |\mathrm{一个}-乙|$，以及统一的先验分布，作为公理。如果两个变量都不知道，$P（X_{\mathrm{2个}}<X_{\mathrm{1个}}）=P（X_{\mathrm{2个}}>X_{\mathrm{1个}}）$。这看起来很明显，因为状态空间$X_{\mathrm{2个}}<X_{\mathrm{1个}}$ 和 $X_{\mathrm{2个}}>X_{\mathrm{1个}}$大小相等。但是，如果${\mathfrak{X}}_{\mathrm{1个}}$ 是已知的 ${\mathfrak{X}}_{\mathrm{2个}}$ 未知，空间中有无限多的数字 $X_{\mathrm{2个}}>{\mathfrak{X}}_{\mathrm{1个}}$ 比 $X_{\mathrm{2个}}<{\mathfrak{X}}_{\mathrm{1个}}$。尽管存在这种不对称性，我们证明$P（X_{\mathrm{2个}}<{\mathfrak{X}}_{\mathrm{1个}}\mid {\mathfrak{X}}_{\mathrm{1个}}）=P（X_{\mathrm{2个}}>{\mathfrak{X}}_{\mathrm{1个}}\mid {\mathfrak{X}}_{\mathrm{1个}}）$， 以便 ${\mathfrak{X}}_{\mathrm{1个}}$ 是的中位数 $p（X_{\mathrm{2个}}|{\mathfrak{X}}_{\mathrm{1个}}）$， 和 ${\mathfrak{X}}_{\mathrm{1个}}$ 在统计上与比率无关 ${\mathfrak{X}}_{\mathrm{2个}}/{\mathfrak{X}}_{\mathrm{1个}}$。我们提出了三个证明，它们适用于一组分布的所有成员。每个成员的区别在于统计模型中隐含的变量（伽玛，高斯等）之间的依存关系形式，但在联合分布中均表现出两个对称性$p（X_{\mathrm{1个}}，X_{\mathrm{2个}}）$ 在没有先验信息的情况下需要具备的条件：变量的可交换性以及边际分布上的非信息先验 $p（X_{\mathrm{1个}}）$ 和 $p（X_{\mathrm{2个}}）$。我们将结论与预测和智能的物理模型相关联，其中已知的“样本”可能是传感器中的当前内部能量，而未知的是其外部感觉原因或未来运动效果中的能量。