OK I'm obviously doing something wrong so some #fediHelp in #probability would help here. Say I have an automaton whose cells can be in any of three states S1, S2, S3 with probability p1, p2, p3 (p1+p2+p3=1). The probability of a cell c changing from S1 to S2 depends on the neighbors being in state S2. c stays in state S1 if it's not “infected” by any of the neighbors. Say p12(c, n) is the probability of c moving from S1 to S2 if n is S2. What's the total probability of c staying in S1?
My initial reasoning has been this: (1 - p2(n)*p12(c, n)) is the probability of n not infecting c. The probability of not being infected is the product over all neighbors of that, times p1(c) (the probability of c being “infectable”). However, making the probability evolve this way gives a very different distribution than actually tracking the state of the cells over multiple runs and then counting how many times a cell gets infected. So what am I doing wrong?
Different approach I'm considering: the probability of c being infected by n is k12(c, n) = p1(c)*p2(n)p12(c, n). So the probability of staying S1 is the complement from all neighbors \prod_n(1 - k12(c, n)), but that can't be, as it can be higher than p1(c). Should it be p1(c)\prod_n(1 - k12(c, n))? But then am I not account for p1(c) too many times? I'm obviously missing something, and being out of my element don't even know where to look things up.
OK I think I'm staring to see why the simulation tracking probabilities is different from the “run n times and compare results”: in the probability-tracking approach, we have no “memory” of the state: the probability of infection propagates “in both directions”, and thus propagates back to the cell that might have triggered the propagation.
Damn. Does this mean that the only way to do this is with the “run multiple times” approach?
To clarify what I mean, imagine the case of a 1D automaton with cells C1, C2, C3. If I run the standard propagation model with C1 initially infected, what happens is that C1 may infect C2, and then when C2 gets infected, it may infect C3. By running this 100 times, I can get an estimate of the probability at every iteration that C2 or C3 are infected.
If I try to propagate probability directly, what happens is that I have initial probability p2(C1) = 1, p2(C2) = 0, p2(C3) = 0.
On the next step, p2^1(C2) = 1 - p12(C2, C1). On the next step p2^2(C3) is computed from the p2^1(C2) … the problem is that THEN C2 has a new infected neighbor (C3) without knowledge that this is actually the “infection” coming from then, so it bounces back. And so on, until they are guaranteed to be infected. I would need a way to avoid this kind of feedback.
And the problem is that this would have to be done in a situation in which initially I don't have a 1 in one cell and 0 elsewhere, but with a situation where I start from nonzero probabilities everywhere. Ideally without tracking where these come from, but this is probably what I cannot avoid.
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