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Philosophy of Science

Negative Probabilities and the Uses of Signed Probability Theory

Edward H. Allen

Published Mar 1, 1976• 22 citations
Publisher

Abstract

The use of negative probabilities is discussed for certain problems in which a stochastic process approach is indicated. An extension of probability theory to include signed (negative and positive) probabilities is outlined and both philosophical and axiomatic examinations of negative probabilities are presented. Finally, a class of applications illustrates the use and implications of signed probability theory.

DOI
10.1086/288669

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The use of negative probabilities is discussed for certain problems in which a stochastic process approach is indicated. An extension of probability theory to include signed (negative and positive) probabilities is outlined and both philosophical and axiomatic examinations of negative probabilities are presented. Finally, a class of applications illustrates the use and implications of signed probability theory.
There are important problems in physics related to the concept of probability. One of these problems is related to negative probabilities used in physics from 1930s. In spite of many demonstrations of usefulness of negative probabilities, physicists looked at them with suspicion trying to avoid this new concept in their theories. The main question that has bothered physicists is mathematical grounding and interpretation of negative probabilities. In this paper, we introduce extended probability as a probability function, which can take both positive and negative values. Defining extended probabilities in an axiomatic way, we show that classical probability is a positive section of extended probability.
This paper is devoted to foundations of probability theory. We discuss interpretations of probability, corresponding mathematical formalisms, and applications to quantum physics. One of the aims of this paper is to show that the probability model based on Kolmogorov's axiomatics cannot describe all stochastic phenomena, i.e., that quantum physics induces natural restrictions of the use of Kolmogorov's theory and that we need to develop non-Kolmogorov models for describing some quantum phenomena. The physical motivations are presented in a clear and brief manner. Thus the reader does not need to have preliminary knowledgeof quantum physics. Our main idea is that we cannot develop non-Kolmogorov models by the formal change of Kolmogorov's axiomatics. We begin with interpretations (classical, frequency, and proportional). Then we present a class of non-Kolmogorov models described by so-called p-adic numbers. Here, in particular, we obtain a quite natural realization of negative probabilities. These negative probability distributions might provide a solution of some quantum paradoxes.
The concept of „negative probability” has become one of interest in research. Initially developed in connection with physics (negative probabilities are important in the analysis of quantum phenomena), nowadays it has applications in many other scientific disciplines. Its main advantage is that it extends the classical probability theory, which involves values between 0 and 1, allowing negative or supraunitary values. From this point of view, negative probabilities are a useful tool that facilitates the calculation and makes analysis more flexible through models. This paper aims to present the theoretical framework, the principles in relation to which the use of negative probabilities in economics, in general, can be understood. The existing literature highlights that the negative probabilities are appropriate in the analysis of stochastic processes and, from this perspective, has developed applications especially in the field of finance. In this sense, negative probabilities are used in models such as the binomial CRR model (The Cox-Ross-Rubinstein Market Model), that evaluate the price of derivatives on the capital market, those that include hidden variable (non-observable) etc. In our analysis we describe these models and, based on them, we try to identify the features of the negative probabilities that can be used in the economic field. The research is a theoretical one, leading the way for new approaches in the direction of using negative probabilities in macroeconomic modelling.
A mathematical framework for observable processes is introduced via the model of systems whose states may be time dependent and described by possibly "negative probabilities". The model generalizes and includes the linearly dependent models or observable operator models for classical discrete stochastic processes. Within this model a general convergence result for finite-dimensional processes, which generalize finite state (hidden) Markov models, is derived. On the philosophical side, the model furthermore offers an explanation for Bell's inequality in quantum mechanics.