Why do we need stochasticity in deterministic simulations?

Question

Assuming the world were deterministic, why would we still need to introduce stochasticity into our simulations?

Answer 1

In a nutshell, to simplify models.

Let’s go with your assumption, even though I don’t believe it. If the universe is completely deterministic, then in any given scenario you choose to model there is one and only one correct answer. Unless you include the complete state space of absolutely everything that determines that answer, your model is wrong. Wrong, wrong, wrong!!!

For instance, if you want to predict how long it will take to fly from New York to London, you need to know the vector sums of all forces acting on the aircraft, which means you need the complete state (down to the atomic level) of the aircraft itself, the passengers, the atmosphere, fluctuations in the magnetic field of the earth, cosmic rays that can trigger upper atmospheric events, etc, etc, ad nauseam. Exclusion of any aspect of the potential forces involved makes your answer wrong.

Clearly, there’s no way to measure it all, and even if there was, there’s no way to maintain so much state information in any computing device we can build. And so we simplify and acknowledge that there is some degree of uncertainty in our model’s predictions/solutions.

When you embrace the existence of uncertainty, it brings us directly to stochastic solutions. One view of probability is that it is a mathematical formalism for modeling uncertainty. Rather than try to model every physical aspect of an aircraft’s flight, we can characterize the likely outcomes based on what proportion of flights require less (or more) than any particular amount of time, i.e., describing the distribution of possible flight times.

Once you adopt distributional modeling, you can see how distributional behaviors propagate though other parts of a system—either analytically, if your system is sufficiently simple, or by generating realizations of the distributions and using replication and sampling via simulation.