Probability distribution as a path and its action integral
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Probability distribution as a path and its action integral Hiroyuki Takeuchi1 Received: 11 June 2019 / Accepted: 4 November 2019 © Japanese Federation of Statistical Science Associations 2019
Abstract To describe the convergence in law of a sequence of probability distributions, “the principle of least action” is introduced nonparametrically into statistics. A probabil‑ ity measure should be treated as a path (in some sense) to apply calculus of varia‑ tions, and it is shown that saddlepoints, which appear in the method of saddlepoint approximations, play a crucial role. An action integral, i.e., a functional of the sad‑ dlepoint, is defined as a definite integral of entropy. As a saddlepoint equation natu‑ rally appears in the Gâteaux derivative of that integral, a unique saddlepoint may be found as an optimal path for this variations problem. Consequently, by virtue of the unique correspondence between probability measures and saddlepoints, the con‑ vergence in law is clearly described by a decreasing sequence of action integrals. Thereby, a new criterion for evaluating the convergence is introduced into statistics and a novel interpretation of saddlepoints is provided. Keywords Action integral · Analytic characteristic function · Calculus of variations · Gâteaux derivative · Principle of least action · Saddlepoint Mathematics Subject Classification 62G20 · 62G30 · 62G99
1 Introduction There are several criteria for the discrepancy between probability distributions, such as the Lévy–Prokhorov metric, the Kolmogorov distance, or the Kull‑ back–Leibler divergence in statistics, probability, or information theory (Amari 1990; Hall 1992; Serfling 1980; Shiryaev 1996). The Edgeworth expansion may also be effective in capturing the difference between a normalized sample mean distribution and the standard normal. Nevertheless, these criteria appear to be arbitrary in a sense, and it is natural to ask whether they can be unified. In this * Hiroyuki Takeuchi [email protected] 1
Department of Economics, Tokyo International University, 13‑1 Matobakita 1‑chome, Kawagoe, Saitama 350‑1197, Japan
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Vol.:(0123456789)
Japanese Journal of Statistics and Data Science
study, a concept of entropy is used as a criterion rather than these distances by introducing “the principle of least action” into statistics. That is, discrepancy will be measured by entropy. It has been widely and successfully used in physics, con‑ trol theory and statistics. Even though there is the likelihood principle in statis‑ tics, it will be applied nonparametrically. Through the action integral, which is defined as a definite integral of Lthe Lagrangian with respect to the time parameter, the convergence in law Fn −→F may be regarded as a decreasing process of it. Furthermore, convergence speed may be clearly evaluated by the asymptotic expansion of that integral. The action integral is a functional, and a stationary point that minimizes it is called an optimal path, if it exists. To define the action integral for a sequence of probability di
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