[8] The Wigner distribution function is routinely used in physics nowadays, and provides the cornerstone of phase-space quantization.
Nevertheless, these regions contribute negatively and crucially to the expected values of observable quantities computed through such distributions.
Xie et al.[12] 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...
[13] The proposed “propensity” concept in Xie et al.[12] 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 [0,1]), but there is no direct information on the station disruption states or their corresponding probabilities.
[15] 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.
[14] Some problems in machine learning use graph- or hypergraph-based formulations having edges assigned with weights, most commonly positive.
Treating a graph weight as a probability of the two vertices to be related is being replaced here with a correlation that of course can be negative or positive equally legitimately.