Bayesian Filtering

Here is the problem -- there is an open state (O) and closed state (C) and a stuck (inactivated) state (S). Transitions between these states happen like this

$O \rightleftharpoons C \rightleftharpoons S$

Transitions between O and S never happen (i.e. the probability is negligible) without C being an intermediary state. The observed current is zero in C and S and one in O. And there is noise. We want to estimate the probability of transitioning back and forth between C and S. Let's say we know everything else (i.e. the probabilities of transitioning back and forth between open and closed states).

First generate some data which looks like.

Click image for full size

The transition probabilities for this plot are

Probability(Stay in Open) = 0.95
Probability(Open --> Closed) = 0.05
Probability(Open --> Stuck/Inactive) = 0.00

Probability(Closed --> Open) = 0.10
Probability(Stay in Closed) = 0.85
Probability(Closed --> Stuck) = 0.05

Probability(Stuck --> Open) = 0.00
Probability(Stuck -- Closed) = 0.003
Probability(Stay in Stuck) = 0.097