Why can’t we track every possible trajectory?
Because the number of possible histories grows every step, and quickly becomes impossible to store. The recursive relationship between conditional probabilities suggests we could model all the trajectories of the simulation at once, and for some simpler systems this is indeed possible, but for most simulations in practice it is computationally infeasible.
Think about how one might store the set of all possible state partition histories for a sequence of coin flips, and then how this can proliferate in time as the simulation advances.
The possible state partition histories grow even though the possible state values of a coin flip always remain the same (heads or tails).
Example: Randomly moving point in 2 dimensions
In practice, for most real-world systems, the set of possible state values also grows. To see how this can be the case, consider the trajectories of a randomly moving point on a simple 2-dimensional grid.
