Probabilistic thinking for simulations

Why do we care?

Probabilities essentially represent sampling all of the possible Trajectories that a simulation could take in Time simultaneously.

Using this representation for a real-world simulation, there are two important use cases.

The first uses all of the possible Trajectories the simulation can take to represent how likely it is to take them. This makes it possible to create algorithms which learn the most likely Parameters for State Partitions to match real-world Data.

The second uses the Probabilities to represent a model in place of the simulation itself. In the right circumstances, this results in algorithms which are much more efficient than sampling multiple Trajectories.

Probabilities and regions

Evaluating the Probability of a particular State Partition History, given Parameters and a Cumulative Timesteps History, looks a similar to the Iterate computation. However, in contrast, this computation does not progress forward in Time.

We might also want to evaluate the Probability of Regions which join together possible values that the whole State Partition History can take.

In many situations, it would be impossible to count the all of the possible values in some Regions, but we can still imagine the computation in this way.

Conditional probabilities

We can relate two successive probability evaluations in Time together by making the answer of the second depend on the outcome of the first.

We call the second of these two evaluations a Conditional Probability because its Probability is conditional on the Probability of the first.

The Probabilities for the whole State Partition History change as the simulation advances in Time by adding the Next State Partition Values into the History.

This concept also applies to the Probability of Regions.

Note how this relationship also describes how the Probabilities of State Partition Histories can evolve in Time. One applies the same calculation to the output from the previous one, and so on, recursively.

So why can’t we just use this recursive relationship to model all the trajectories of the simulation at once?

For some simpler systems this is indeed possible, but for most simulations in practice this is computationally infeasible.

Think about how one might store the set of all Possible State Partition Histories on a computer, and then how this can proliferate in Time as the simulation advances. The memory required tends to exponentially increase.

To be continued…