在有限证据的基础上对连续统假设的一致推理
Consistent reasoning about a continuum of hypotheses on the basis of finite evidence
摘要：
在现代贝叶斯看来，经典概率论只是传统逻辑的扩展，即，一种定量工具，允许在存在不确定性时进行一致的推理。然而，经典理论的前提是--至少在原则上--实验者可以积累的证据量总是与假设空间的大小相匹配。我调查的框架一致的推理必须修改在非经典的情况下，假设形成一个连续体，但最大的证据，通过实验是不允许超过一些有限的上限。除了基本的一致性要求有关的准备和组成的系统，以及概率的连续性，我表明，修改后的理论必须有一个内部对称同构的酉群。这样看来，在有限证据的基础上对一系列假设进行合理推理的唯一一致的算法是由复希尔伯特空间中的量子理论提供的。
英文摘要：
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.
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