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AIP conference proceedings

Bayesian versus Frequentist Predictions in Quantum Tomography

Rüdiger Schack

Published Jan 1, 2007• 2 citations
Publisher

Abstract

This short paper is a reply to Andrei Khrennikov’s challenge to give a concrete quantum‐mechanical example where the Bayesian and frequentist approaches to probability lead to different predictions. The paper first reviews the difference between the Bayesian and frequentist approaches to hypothesis testing and then presents a simple example in quantum tomography for which Bayesian and frequentist predictions differ.

DOI
10.1063/1.2713461

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This short paper is a reply to Andrei Khrennikov’s challenge to give a concrete quantum‐mechanical example where the Bayesian and frequentist approaches to probability lead to different predictions. The paper first reviews the difference between the Bayesian and frequentist approaches to hypothesis testing and then presents a simple example in quantum tomography for which Bayesian and frequentist predictions differ.
Quantum mechanics is a fundamentally probabilistic theory (at least so far as the empirical predictions are concerned). It follows that, if one wants to properly understand quantum mechanics, it is essential to clearly understand the meaning of probability statements. The interpretation of probability has excited nearly as much philosophical controversy as the interpretation of quantum mechanics. 20th century physicists have mostly adopted a frequentist conception. In this paper it is argued that we ought, instead, to adopt a logical or Bayesian conception. The paper includes a comparison of the orthodox and Bayesian theories of statistical inference. It concludes with a few remarks concerning the implications for the concept of physical reality.
Physicists have, hitherto, mostly adopted a frequentist conception of probability, according to which probability statements apply only to ensembles. It is argued that we should, instead, adopt an epistemic, or Bayesian conception, in which probabilities are conceived as logical constructs rather than physical realities, and in which probability statements do apply directly to individual events. The question is closely related to the disagreement between the orthodox school of statistical thought and the Bayesian school. It has important technical implications (it makes a difference, what statistical methodology one adopts). It may also have important implications for the interpretation of the quantum state. 1 1.
Physicists have, hitherto, mostly adopted a frequentist conception of probability, according to which probability statements apply only to ensembles. It is argued that we should, instead, adopt an epistemic, or Bayesian, conception, in which probabilities are conceived as logical constructs rather than physical realities and in which probability statements do apply directly to individual events. The question is closely related to the disagreement between the orthodox school of statistical thought and the Bayesian school. It has important technical implications (it makes a difference what statistical methodology one adopts). It may also have important implications for the interpretation of the quantum state.
Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cramér-Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cramér-Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.