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Behavioral and Brain Sciences

Does quantum uncertainty have a place in everyday applied statistics?

Andrew Gelman, Michael Betancourt

Published May 14, 2013• 8 citations
Publisher

Abstract

We are sympathetic to the general ideas presented in the article by Pothos & Busemeyer (P&B): Heisenberg's uncertainty principle seems naturally relevant in the social and behavioral sciences, in which measurements can affect the people being studied. We propose that the best approach for developing quantum probability models in the social and behavioral sciences is not by directly using the complex probability-amplitude formulation proposed in the article, but rather, more generally, to consider marginal probabilities that need not be averages over conditionals.

DOI
10.1017/s0140525x12002944

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We are sympathetic to the general ideas presented in the article by Pothos & Busemeyer (P&B): Heisenberg's uncertainty principle seems naturally relevant in the social and behavioral sciences, in which measurements can affect the people being studied. We propose that the best approach for developing quantum probability models in the social and behavioral sciences is not by directly using the complex probability-amplitude formulation proposed in the article, but rather, more generally, to consider marginal probabilities that need not be averages over conditionals.
The three main points of this article are: 1. Quantum mechanical data differ from conventional data: for example, joint distributions usually cannot be defined conventionally; 2. rigorous methods have been developed for analyzing such data; the methods often use quantum-consistent analogs of classical statistical procedures; 3. with these procedures, statisticians, both data-analytic and more theoretically oriented, can become active participants in many new and emerging areas of science and biotechnology. In the physical realm described by quantum mechanics, many conventional statistical and probabilistic assumptions no longer hold. Probabilistic ideas are central to quantum theory but the standard Kolmogorov axioms are not uniformly applicable. Studying such phenomena requires an altered model for sample spaces, for random variables and for inference and decision making. The appropriate decision theory has been in development since the mid-1960s. It is both mathematically and statistically rigorous and conforms to the requirements of the known physical results. This article provides a tour of the structure and current applications of quantum-consistent statistical inference and decision theory. It presents examples, outlines the theory and considers applications and open problems. Certain central concepts of quantum theory are more clearly apprehended in terms of the quantum-consistent statistical decision theory. For example, the Heisenberg uncertainty principle can be obtained as a consequence of the quantum version of the Cramer-Rao inequality. This places concepts of statistical estimation and decision theory, and thus the statistician, at the center of the quantum measurement process. Quantum statistical inference offers considerable scope for participation by the statistical community, in both applications and foundational questions.
Can one really predict something new and interesting using quantum theory in the social or behavioral sciences? Consider the law of reciprocity described in Chapter 2: according to this law, the probability of transiting from one state to another is equal to the probability of making this transition in the opposite direction. Now this symmetrical property is a pretty bold prediction to make in the social and behavioral sciences. Does this really work? To answer this question, in this chapter we empirically test this law by applying the quantum principles to an important empirical problem concerning the effects of question order on attitude judgments. For example, suppose you are asked “How happy are you with life in general?” This question could be preceded or followed by the question “How happy are you with your marriage?” If the marriage question comes first, then the happiness rating for the general life question tends to be substantially depressed, producing a large order effect (Tourangeau et al ., 1991). Quantum physics was originally developed to understand how measurement affects the system under investigation, which led to Heisenberg's famous uncertainty principle. The potential reactivity to measurement by the person being measured has an even longer history in psychology (Tourangeau et al ., 2000). The effects of question order have been an important issue to survey researchers interested in studying beliefs and attitudes (Schuman & Presser, 1981). Measurement of a belief or attitude directs attention to a subset of a person's knowledge, which is then used in guiding subsequent judgments.