What is noise?
Let us consider the throw of a dice.
It is a very “complex” physical phenomenon. Complex in the sense that many factors ought to be taken into account as, for instance, initial forces, gravity, aerodynamics, bouncing forces, and so forth. Initial forces themselves depend on the musculature, the shape, the health of the body, on the roughness of the skin, and precise shape of the hands and fingers, on the motor control and even “will” of the thrower. Gravity depends on the balance of the dice and the main gravity attraction of earth, but also on local perturbations due to geological constraints and eventually the presence of heavy objects in the vicinity. It also depends on far away attractions as, to cite only the most preeminent one, the position of the moon at the time of the throw. Aerodynamics depend on anything that may generate a breath of air, from the neighbor’s sneeze to the butterfly effect on the local climate.
Is the trajectory of the dice a deterministic physical process? A very argued question but absolutely not relevant to our problem. Would it be completely deterministic, it will not help us to predict the outcome as, obviously, we lack boundless information to be able to reconstruct the course of events.
If the dice is “loaded,” for instance, toward an outcome of 6 and if we consider only a standard vertical gravity, we may predict the probability of this outcome. Any deviation from this prediction can be interpreted as “noise.” However, it is not related to any physical property of the system, this deviation is only the result of our ignorance (or simplification choices) that makes us discard all the relevant information enumerated above.
To ensure that a throw is “honest”, you withdraw any observable information using, for instance, a balanced dice that prevents you from using any macroscopic gravity prediction and a cup that deprives you from any reachable information about the initial forces. “Noise” is then so preeminent, your ignorance is so complete, that the best prediction you can reach is a uniform distribution on the six possible outcomes. The “honesty” of the dice is nothing but your own “ignorance”.