## On Stochastic Processes

Why can't I shake the feeling that a stochastic process really... isn't? In previous posts we have been able to frame a deterministic system in terms of its Markov transform matrix, and going backward really doesn't seem like a problem. So what's going on, I can't help but wonder? Are deterministic systems a subclass of random processes? Are they isomorphic spaces, or convergent at the limit as we take smaller pieces of time? Was Einstein right, that it all converges to a determined state, and "God does not roll dice?" What do you think?

With your most recent post, I now better understand (a little) what you are asking. So I'll take a swing at this now, but my answer might border on Philosophy. :-)

If a stochastic process has only one end state, and you only observe the beginning and end states, then it is indistinguishable from a deterministic system. In this case is is a subclass, but maybe a trivial one.

If other end states exist, it would be a mistake to assume the end state you observe is the only end state possible. Now this seems obvious when you can write down all the intermediate states, but in practice people tend to discount this sort of information, and will believe what they observe is the only outcome possible, or that this particular outcome is much more likely than it really is/was. Sometimes this is called "hindsight bias".

Therefore, IMO, "God rolls dice", but we only get to see how they have fallen.