A Stochastic Model and its Epistemological Inferences

Eugene C. Barrows

Matthew Ho, Mile Gu, Thomas J. Elliott

A growing body of work has established the modeling of stochastic processes as a promising area of application for quantum technologies; it has been shown that quantum models are able to replicate the future statistics of a stochastic process while retaining less information about the past than any classical model must, even for a purely classical process. Such memory-efficient models open a potential future route to study complex systems in greater detail than ever before and suggest profound consequences for our notions of structure in their dynamics. Yet, to date methods for constructing these quantum models are based on having a prior knowledge of the optimal classical model. Here, we introduce a protocol for blind inference of the memory structure of quantum models---tailored to take advantage of quantum features---direct from time-series data, in the process highlighting the robustness of their structure to noise. This in turn provides a way to construct memory-efficient quantum models of stochastic processes while circumventing certain drawbacks that manifest solely as a result of classical information processing in classical inference protocols.

Inge Svein Helland

Every experiment or observational study is made in a context. This context is being explicitly considered in this book. To do so, a conceptual variable is defined as any variable which can be defined by (a group of) researchers in a given setting. Such variables are classified. Sufficiency and ancillarity are defined conditionally on the context. The conditionality principle, the sufficiency principle and the likelihood principle are generalized, and a tentative rule for when one should not condition on an ancillary is motivated by examples. The theory is illustrated by the case where a nuisance parameter is a part of the context, and for this case, model reduction is motivated. Model reduction is discussed in general from the point of view that there exists a mathematical group acting upon the parameter space. It is shown that a natural extension of this discussion also gives a conceptual basis from which essential parts of the formalism of quantum mechanics can be derived. This implies an epistemological basis for quantum theory, a kind of basis that has also been advocated by part of the quantum foundation community in recent years. Born's celebrated formula is shown to follow from a focused version of the likelihood principle together with some reasonable assumptions on rationality connected to experimental evidence. Some statistical consequences of Born's formula are sketched. The questions around Bell's inequality are approached by using the conditionality principle for each observer. The objective aspects of the world are identified with the ideal inference results upon which all observers agree (epistemological objectivity).

A. O. Bolivar

Tsubasa Ichikawa

Abstract Logical inference leads to one of the major interpretations of probability theory called logical interpretation, in which the probability is seen as a measure of the plausibility of a logical statement under incomplete information. Assuming that our usual inference procedure is working rationally for every set of logical propositions represented in terms of commuting projectors on a given Hilbert space, we extend the logical interpretation to quantum mechanics and derive the Born rule. Our result implies that, from epistemological viewpoints, we can regard quantum mechanics as a natural extension of classical probability theory.