Szekely provides a convincing argument of negative probability here: http://www.wilmott.com/pdfs/100609_gjs.pdf
What does a reformulation of classical information theory built from negative probabilities mean?
I am looking for applications to entropy and communications theory as well.
In the context of information theory, negative probabilities explain the quantum mechanical phenomenon of probability backflow, as described in section 5 of Waiting for the Quantum Bus (Bracken and Melloy, 2014).
Probability backflow: You wait by the road in the hope of flagging down the bus if and when it reaches you from the left. In classical mechanics, the longer you wait, the more doubtful you become that the bus is still on its way. In quantum mechanics, despite the fact that the bus is definitely travelling from left to right, the probability of finding it to your left on measuring its position may increase as the time of measurement increases.
To interpret this in terms of the flow of probability, we may say that all the probability moves to the right with the motion of the quantum bus, just as in the case of the classical bus, but now not all that probability is positive. Negative probability moving to the right has the same effect on the total probabilities as positive probability moving to the left, thus giving rise to the backflow phenomenon.
