Graph entropy 8: Negative probabilities

After some introductory posts defining 4 generalized entropies, named H₁ to H₄, and using the last two of them to define a Tree-Graph Entropy (that you should had read first, starting here) I will try now to extend the usage of entropies H₃ and H₄ to the unknown realm of... negative probabilites!

Negative probabilities 

But wait... what the point is in even considering that negative probabilities are a "thing"? What they do represent, if anything, in a real world case?

Good question! Well, imagine the example of the car engines, we designed our eperiment to detect gas and electric engines, but we could had found cars using  a hidrogen engine too! If it hapends, we would end up with and impossible event -finding a car with a totally different kind of engine- having a real probability.

So let's say a 1% of the cars were found to use an "alien" hidrogen engine. This new pi = 0.01 is not adding information to our experiment, in fact, it is proven us wrong in our assumptions, so it is decreasing our -pretended- knowledge: it is acting as a negative entropy, so I decided to try using it as negative probability, assigning it the value p = -0.01

It really makes no sense to just add a negative sign and pretend entropy will be lower, the original formula does not allow for negative probabilities, so a little fix was needed, not much really:

H₂(P) = ∏(2-|p_i|^p_i) 

If we now plot both entropies generators in the new extended range (-1, 1), this is what you get:

 

So, when a negative probability is added to the formula, you are multiplying by a coeficient smaller that 1, so lowering the resulting product entropy, as intended!

In fact, for p=-0.01, the added term in the product is 0.952871, meaning the H₂ entropy of the experiment results will be 0.471285% lower than initially expected, what makes perfect sense to me.

So what if those two entropies can handle this case? Not sure if it could be of any use for some real experimental setup, but it is really funny to play around with those ideas, so I did (and, should you find a proper case for this, I am open to collaboration!).