Modeling with Negative Probabilities? Background:
As I understand the role of probability in Quantum Mechanics, the idea is that no observable event can have negative probability, but that it can make sense for unobserved quantities to have negative probability, so long as when marginalizing over unobserved events we obtain a proper probability distribution over observable ones.
Bayesian hierarchical models often contain unobservable latent variables. Conceivably, endowing some of these latent variables negative probability (densities) could still result in a well defined probability density over the observable nodes. And, also conceivably, doing so would allow for a more parsimonious description of the marginal probability distribution than is possible using latent variables with positive probabilities only.
More Background: Maybe an example is in order. Let's consider the standard two-cluster Gaussian mixture model with unknown mean and identical, known variance in dimension 1, for now with positive probabilities as usual:
$$z \sim bern(\rho)$$
$$y|z=0 \sim N(\mu_0,1)$$
$$y|z=1 \sim N(\mu_1,1)$$
Here, the unobserved latent variable is $z$, and when marginilizing over it, we are left with:
$$\delta_y(y) = \rho \delta_{N(\mu_0,1)}(y) + (1-\rho) \delta_{N(\mu_1,1)}(y)$$
where $\delta_y$ is the density of $y$ and $\delta_{N(a,b)}$ gives the density of a normal distribution with mean $a$ and variance $b$.
Here's what that density looks like for $\mu_0=1$,$\mu_1=-1$ and $\rho=0.9$:

(Since the peaks are close relative to the standard error it looks much like a single normal distribution).
Now I'm going to change the second density from being a $N(\mu_1,1)$ to a distribution with negative probabilities by changing it's density to this function:
$$\delta_{\aleph} := \delta_{N(\mu_1,0)}\textrm{sgn}(\mu_1-y)$$
where $\textrm{sgn}(a)$ is the sign function, so it looks like this:

If we just plug this into the previous expression for the marginal density, we get:
$$\delta_{-}(y) = \rho \delta_{N(\mu_0,1)}(y) + (1-\rho) \delta_{\aleph}(y)$$
which looks like this

This is by all accounts a proper marginal: if I had thought of it, I could happily use it as a likelihood for my data without any need for negative probabilities by directly applying this distribution.
Question: Are there examples of negative densities being used in the context of latent variable models as a modeling device in order to define a generative model of observed data?
Such a thing exists in the context of the outcome of quantum phenomena, such as the double slit experiment. See this Wikipedia article for more on negative probabilities.

The negative probabilities can have different interpretations.

*

*In calculations we may have negative coefficients.
In the comments Whuber gives an example of an expression for the probability density function of a sum of gamma distributed variables. This density can be seen as a mixture of gamma distribution where some terms are negative.
A simpler example is the computation of the probability $$P(\text{$A$ or $B$}) = P(\text{$A$}) + P(\text{$B$}) - P(\text{$A$ and $B$})$$ here there is also a negative coefficient and a negative term $ - P(\text{$A$ and $B$})$, but there is no event whose probability is negative. We are just making a computation where we subtract a probability.


*The events themselves have negative probability in computations. This is what the slit experiment argues. There is the similar formula $P(\text{$A$ or $B$}) = P(\text{$A$}) + P(\text{$B$}) - P(\text{$A$ and $B$})$ but what they compute is $P(\text{$A$ and $B$})$ and they find out that this term is negative.
There can be different ways how these negative probabilities occur

*

*As probabilities of events in a virtual model. Those events are never observed themselves and may not even be real and are only our interpretation of the underlying mathematical mechanics of physics.
In quantum mechanics these negative probabilities are not paradoxical because they are pseudo-probabilities. The reality –what we observe – is the only place where probability makes sense. Whatever mathematics and physical description is underlying the observations, it is not 'real' and interpretations of probability are irrelevant in that realm.
For example, in the double slit experiment the negative probability only arrises because we like to think of a non observable event (the photon goes through both slits) and attach a probability to this. But this idea of a photon going through both slits is an imagination and is only how we interpret the unobservable world behind the observable world. It is an atomic/physical interpretation of a subatomic/metaphysical world.


*A negative probability function can also arrise to simplify computation or because we use approximations that might become negative. In these cases the mathematics is not physical and only happens to be negative due to the use of a model.
The above gives some ways how negative probability may arrise but in all cases it does not mean that there is really some event that has negative probability. The 'negative probability' is a side-effect from using mathematical expression and falsely interpreting terms in those expressions as correct expressions of probability for some real event. The negative may occur because either the expression is not correct (e.g. approximation or simplification) or because the interpretation of the term as an event is not real (e.g. metaphysical interpretations of a quantum mechanical world with actual point particles passing through two slits simultaneously).
