As a roboticist, implementing visual odometry on a robot, homogeneous coordinates are convenient for projections of a non-moving object on an image sensor at $t$ and $t+1$ to estimate its position, see for example the "Fundamental matrix".
Homogeneous coordinates add an additional element to for example a 2D point that can just be seen as a scaling factor of the elements before. When I was contemplating message passing and probabilities I suddenly realized that the same phenomenon is true there. A probability of cardinality $2$ can be represented by $[0.2, 0.8]$ or by $[0.2, 0.8, 1.0]$ where the last component automatically keeps track of the normalization required to keep it a proper probability. If it is multiplied by a scalar, the underlying probability does not change.
I am not an expert in any way, so my laymen question is: How is this "representation" called in probability theory?