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The probability of the outcome of an experiment is never negative, although a quasiprobability distribution allows a negative probability, or quasiprobability for some events. These distributions may apply to unobservable events or conditional probabilities.

Physics and mathematics

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, negative money is valid. 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 random variables, Y and Z, with ordinary (not signed / not quasi) distributions such that X, Y are independent and X + Y = Z in distribution. Thus X can always be interpreted as the "difference" of two ordinary random variables, Z and Y. If Y is interpreted as a measurement error of X and the observed value is Z then the negative regions of the distribution of X are masked / shielded by the error 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. For this reason, it has later been better known as the Wigner quasiprobability 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 quasiprobability distribution are highly constrained, and prevent direct measurability of the negative regions of the distribution. Nevertheless, these regions contribute negatively and crucially to the expected values of observable quantities computed through such distributions.

Engineering

The concept of negative probabilities has also been proposed for reliable facility location models where facilities are subject to negatively correlated disruption risks when facility locations, customer allocation, and backup service plans are determined simultaneously. Li et al. proposed a virtual station structure that transforms a facility network with positively correlated disruptions into an equivalent one with added virtual supporting stations, and these virtual stations were subject to independent disruptions. This approach reduces a problem from one with correlated disruptions to one without. Xie et al. later showed how negatively correlated disruptions can also be addressed by the same modeling framework, except that a virtual supporting station now may be disrupted with a “failure propensity” which

... inherits all mathematical characteristics and properties of a failure probability except that we allow it to be larger than 1...

This finding paves ways for using compact mixed-integer mathematical programs to optimally design reliable location of service facilities under site-dependent and positive/negative/mixed facility disruption correlations.

The proposed “propensity” concept in Xie et al. turns out to be what Feynman and others referred to as “quasi-probability.” Note that when a quasi-probability is larger than 1, then 1 minus this value gives a negative probability. In the reliable facility location context, the truly physically verifiable observation is the facility disruption states (whose probabilities are ensured to be within the conventional range ), but there is no direct information on the station disruption states or their corresponding probabilities. Hence the disruption "probabilities" of the stations, interpreted as “probabilities of imagined intermediary states,” could exceed unity, and thus are referred to as quasi-probabilities.

See also

• Existence of states of negative norm (or fields with the wrong sign of the kinetic term, such as Pauli–Villars ghosts) allows the probabilities to be negative. See Ghosts (physics).
• Signed measure
• Wigner quasiprobability distribution

References

1. Dirac, P. A. M. (1942). "Bakerian Lecture. The Physical Interpretation of Quantum Mechanics". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 180 (980): 1–39. Bibcode:1942RSPSA.180....1D. doi:10.1098/rspa.1942.0023. JSTOR 97777.
2. Feynman, Richard P. (1987). "Negative Probability" (PDF). In Peat, F. David; Hiley, Basil (eds.). Quantum Implications: Essays in Honour of David Bohm. Routledge & Kegan Paul Ltd. pp. 235–248. ISBN 978-0415069601.
3. Khrennikov, Andrei Y. (March 7, 2013). Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Springer Science & Business Media. ISBN 978-94-009-1483-4.
4. Székely, G.J. (July 2005). "Half of a Coin: Negative Probabilities" (PDF). Wilmott Magazine: 66–68. Archived from the original (PDF) on 2013-11-08.
5. Ruzsa, Imre Z.; SzéKely, Gábor J. (1983). "Convolution quotients of nonnegative functions". Monatshefte für Mathematik. 95 (3): 235–239. doi:10.1007/BF01352002. S2CID 122858460.
6. Ruzsa, I.Z.; Székely, G.J. (1988). Algebraic Probability Theory. New York: Wiley. ISBN 0-471-91803-2.
7. Wigner, E. (1932). "On the Quantum Correction for Thermodynamic Equilibrium". Physical Review. 40 (5): 749–759. Bibcode:1932PhRv...40..749W. doi:10.1103/PhysRev.40.749. hdl:10338.dmlcz/141466.
8. Bartlett, M. S. (1945). "Negative Probability". Mathematical Proceedings of the Cambridge Philosophical Society. 41 (1): 71–73. Bibcode:1945PCPS...41...71B. doi:10.1017/S0305004100022398. S2CID 12149669.
9. Snyder, L.V.; Daskin, M.S. (2005). "Reliability Models for Facility Location: The Expected Failure Cost Case". Transportation Science. 39 (3): 400–416. CiteSeerX 10.1.1.1.7162. doi:10.1287/trsc.1040.0107.
10. Cui, T.; Ouyang, Y.; Shen, Z-J. M. (2010). "Reliable Facility Location Design Under the Risk of Disruptions". Operations Research. 58 (4): 998–1011. CiteSeerX 10.1.1.367.3741. doi:10.1287/opre.1090.0801. S2CID 6236098.
11. Li, X.; Ouyang, Y.; Peng, F. (2013). "A supporting station model for reliable infrastructure location design under interdependent disruptions". Transportation Research Part E. 60: 80–93. doi:10.1016/j.tre.2013.06.005.
12. ^ Xie, S.; Li, X.; Ouyang, Y. (2015). "Decomposition of general facility disruption correlations via augmentation of virtual supporting stations". Transportation Research Part B. 80: 64–81. doi:10.1016/j.trb.2015.06.006.
13. Xie, Siyang; An, Kun; Ouyang, Yanfeng (2019). "Planning facility location under generally correlated facility disruptions: Use of supporting stations and quasi-probabilities". Transportation Research Part B: Methodological. 122. Elsevier BV: 115–139. doi:10.1016/j.trb.2019.02.001. ISSN 0191-2615.

• Quantum mechanics
• Exotic probabilities
• Mathematical finance