I was very intrigued by the answer here. I would like to have a more layman explanation of what negative probabilities could mean and their applications, possibly with examples. For instance, what would it mean for an event to have probability -10%, according to these extended measures of probability?

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    $\begingroup$ Vague memories of Dick Feynman riffing on this topic... on a little further reflection, I think he was riffing on making new ways of thinking about things, and using the example of negative variances. $\endgroup$ – Alexis Mar 15 '18 at 15:57
  • $\begingroup$ In layman's terms, 1% chance means "very unlikely". 0% chance is "no way". Minus 10% is like "you gots to be kidding" $\endgroup$ – Mark L. Stone Apr 11 '18 at 16:12

Negative probabilities are essentially related to Quantum Mechanics and it's still hard to find meaningful examples outside of it. Formally negative probability is just using a signed measure instead of a measure for $P$. But actually, many intuitive concepts of probabilities become ill-defined. For example, sampling from a signed distribution is something that does not make any practical sense (in a non quantum reality).

In QM, negative probabilities are about properties that can't be observed, only hypothesized: like the electron is at position $x$ with speed $p$ (see Wigner quasi distribution). The signed probabilities about unobservable events in turn explain the probabilities of observable events. But in the end, the observable events have positive probabilities so that nothing about negative probabilities can directly be observed. They only play a role in the calculation. In other words (Richard Feynman's):

It is not my intention here to contend that the final probability of a verifiable physical event can be negative. On the other hand, conditional probabilities and probabilities of imagined intermediary states may be negative in a calculation of probabilities of physical events or states. If a physical theory for calculating probabilities yields a negative probability for a given situation under certain assumed conditions, we need not conclude the theory is incorrect. Two other possibilities of interpretation exist. One is that the conditions (for example, initial conditions) may not be capable of being realized in the physical world. The other possibility is that the situation for which the probability appears to be negative is not one that can be verified directly. A combination of these two, limitation of verifiability and freedom in initial conditions, may also be a solution to the apparent difficulty.

Negative probabilities revolve around the idea of cancellation. In classical probabilities, when an event has occurred, it has occurred and there is nothing you can change about it. In negative probabilities events can be cancelled. There are positive events and negative (anti) events. Each negative event ("I saw an anti-chicken") merges with a corresponding positive event ("I saw a chicken") by cancelling it. But what if a negative event finds no positive event to cancel? Would you observe the negative event? The fact is the question does not happen in QM: any observable event has a positive probability. Chickens and anti-chickens are actually both perfectly invisible.

Somehow, for the moment, negative probabilities ask more questions than they provide answers because of their highly anti-intuitive nature. Fundamentally, since signed probability are a formulation of QM, and that QM has fundamentally no simple "down to earth" interpretation, it is very unlikely signed probabilities will ever have a "down to earth" interpretation. I'm quoting this from the answer you pointed:

As a final remark, whuber is absolutely right that it isn't really legal to call anything a probability that doesn't lie in [0,1], at the very least, not for the time being. Given that "negative probabilities" have been around for so long I don't see this changing in the near future, not without some kind of colossal breakthrough.

  • $\begingroup$ I think it's worth pointing out more prominently that in the standard mathematical apparatus of QM there are no negative probabilities. There are complex amplitudes, but probabilities are given by the squared absolute values of these amplitudes, i.e. are always non-negative. $\endgroup$ – amoeba says Reinstate Monica Apr 17 '18 at 13:37

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