# Negative probability

﻿
Negative probability

In 1942, Paul Dirac wrote a paper "The Physical Interpretation of Quantum Mechanics" where he introduced the concept of negative energies and negative probabilities:

"Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money."

The idea of negative probabilities later received increased attention in physics and particularly in quantum mechanics. Richard Feynman argued that no one objects to using negative numbers in calculations, although "minus three apples" is not a valid concept in real life. Similarly he argued how negative probabilities as well as probabilities above unity possibly could be useful in probability calculations.

Negative probabilities have later been suggested to solve several problems and paradoxes. Half-coins provide simple examples for negative probabilities. These strange coins were introduced in 2005 by Gábor J. Székely.  Half-coins have infinitely many sides numbered with 0,1,2,... and the positive even numbers are taken with negative probabilities. Two half-coins make a complete coin in the sense that if we flip two half-coins then the sum of the outcomes is 0 or 1 with probability 1/2 as if we simply flipped a fair coin.

In Convolution quotients of nonnegative definite functions and Algebraic Probability Theory  Imre Z. Ruzsa and Gábor J. Székely proved that if a random variable X has a signed or quasi distribution where some of the probabilities are negative then one can always find two other independent random variables, Y, Z, with ordinary (not signed / not quasi) distributions such that X + Y = Z in distribution thus X can always be interpreted as the `difference' of two ordinary random variables, Z and Y.

Another example known as the Wigner distribution in phase space, introduced by Eugene Wigner in 1932 to study quantum corrections, often leads to negative probabilities, or as some would say "quasi-probabilities". For this reason, it has later been better known as the Wigner quasi-probability distribution. In 1945, M. S. Bartlett worked out the mathematical and logical consistency of such negative valuedness. The Wigner distribution function is routinely used in physics nowadays, and provides the cornerstone of phase-space quantization. Its negative features are an asset to the formalism, and often indicate quantum interference. The negative regions of the distribution are shielded from direct observation by the quantum uncertainty principle: typically, the moments of such a non-positive-semidefinite quasi-probability distribution are highly constrained, and prevent direct measurability of the negative regions of the distribution. But these regions contribute negatively and crucially to the expected values of observable quantities computed through such distributions, nevertheless.

Negative probabilities have more recently been applied to mathematical finance. In quantitative finance most probabilities are not real probabilities but pseudo probabilities, often what is known as risk neutral probabilities. These are not real probabilities, but theoretical "probabilities" under a series of assumptions that helps simplify calculations by allowing such pseudo probabilities to be negative in certain cases as first pointed out by Haug in 2004 .

A rigorous mathematical definition of negative probabilities and their properties was recently derived by Mark Burgin and Gunter Meissner (2011). The authors also show how negative probabilities can be applied to financial option pricing.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Probability — is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the… …   Wikipedia

• Probability interpretations — The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it… …   Wikipedia

• Probability space — This article is about mathematical term. For the novel, see Probability Space (novel). In probability theory, a probability space or a probability triple is a mathematical construct that models a real world process (or experiment ) consisting of… …   Wikipedia

• Negative binomial distribution — Probability mass function The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation. notation: parameters: r > 0 number of failures until the experiment is stopped (integer,… …   Wikipedia

• Probability theory — is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non deterministic events or measured… …   Wikipedia

• Probability matching — is a suboptimal decision strategy in which predictions of class membership are proportional to the class base rates. Thus, if in the training set positive examples are observed 60% of the time, and negative examples are observed 40% of the time,… …   Wikipedia

• negative reinforcement — noun (behaviourism) In conditioning situations, a stimulus, usu aversive, that increases the probability of escape or avoidance behaviour • • • Main Entry: ↑negate …   Useful english dictionary

• Probability distribution — This article is about probability distribution. For generalized functions in mathematical analysis, see Distribution (mathematics). For other uses, see Distribution (disambiguation). In probability theory, a probability mass, probability density …   Wikipedia

• Probability-generating function — In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are… …   Wikipedia

• probability theory — Math., Statistics. the theory of analyzing and making statements concerning the probability of the occurrence of uncertain events. Cf. probability (def. 4). [1830 40] * * * Branch of mathematics that deals with analysis of random events.… …   Universalium