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Journal of Physics Conference Series

Toward a probabilistic theory of the measuring systems

Giovanni Battista Rossi, Francesco Crenna, Alice Palazzo

Published Aug 1, 2018• 0 citations
Publisher

Abstract

In the current debate on measurement science, there is growing interest in identifying some generic framework for the modelling of the measurement system. Recently, we have proposed sufficient conditions for an empirical system to qualify as a measuring system, and have proposed a deterministic framework based on that. Here we consider turning that framework into a corresponding probabilistic one, in other to properly accounting for measurement uncertainty.

DOI
10.1088/1742-6596/1065/7/072008

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In the current debate on measurement science, there is growing interest in identifying some generic framework for the modelling of the measurement system. Recently, we have proposed sufficient conditions for an empirical system to qualify as a measuring system, and have proposed a deterministic framework based on that. Here we consider turning that framework into a corresponding probabilistic one, in other to properly accounting for measurement uncertainty.
In the representational approach, measurement is defined on the basis of primitive (in the context) notion of empirical relation. To account for measurement uncertainty, it is quite natural to consider probabilistic (or, more generally, non deterministic) empirical relations. Here the notion of probabilistic relation is investigated. After a brief historical outline, we point out that probabilistic relations appear in everyday measurement situation. To demonstrate this, a simple illustrative experiment has been developed, whose results are presented and discussed.
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Debates about "uncertainty in mathematics" and "the probabilistic turn in physics" routinely conflate three things that should be kept apart: the internal deductive relation between axioms and theorems within a formal system (which I call L-m), the approximate fit between a formal model and its target system (L-model), and the empirical or quantum probabilities of measurement outcomes (L-measure). I argue that sliding a probability or uncertainty predicate from one layer to another is a Rylean category mistake, and I supply an operational diagnostic — "probable with respect to what?" — that exposes such slides. The claim is deliberately deflationary: I do not propose a new interpretation of probability, a new measure of uncertainty, or a foundationalist defence of mathematical certainty. Rather, I offer a heuristic taxonomy whose payoff is diagnostic. I apply it to two canonical worked examples — neutrino oscillation and the double-slit experiment — and to four live objections: that the mathematics of quantum mechanics is itself probabilistic, that QBism locates probability in the agent's credence, that Gödelian incompleteness and non-standard models make mathematics itself uncertain, and that the whole distinction is merely terminological. The framework is interpretation-neutral: however one reads probability (frequentist, Bayesian, propensity, QBist), the question "probable with respect to what?" still has to be answered, and most intractable "uncertainty in mathematics" disputes dissolve once it is. The paper closes by stating the falsification conditions under which the taxonomy would lose content.
Ontological models are attempts to quantitatively describe the results of a probabilistic theory, such as Quantum Mechanics, in a framework exhibiting an explicit realism-based underpinning. Unlike either the well known quasi-probability representations, or the "r-p" vector formalism, these models are contextual and by definition only involve positive probability distributions (and indicator functions). In this article we study how the ontological model formalism can be used to describe arbitrary statistics of a system subjected to a finite set of preparations and measurements. We present three models which can describe any such empirical data and then discuss how to turn an indeterministic model into a deterministic one. This raises the issue of how such models manifest contextuality, and we provide an explicit example to demonstrate this. In the second half of the paper we consider the issue of finding ontological models with as few ontic states as possible.