Interpretation of the first passage time

[phyalan]

Hi all,
Suppose I have a kinetic model for a stochastic system of three states, as shown in the attachment. I solve for the probability distribution of the first passage time from A to B and I get the distribution shown on the right hand side.

I can understand that if there is a peak in the distribution, we can say there is some most probable first time(fpt) for the system to transit from A to B because in the path A->C->B, one can go back and fourth between A and C before reaching B. But how about the non-zero peak at t=0 in the distribution? I know it comes from the path A->B because this path has no intermediate stop, the distribution follows a single exponential function but I am confused about how to interpret it physically. Does it means that the system 'typically' takes 0 time to transit to B in this path?

And if I want to have a estimation of the most probable fpt, is taking the weighted mean of the two peaks with respect to their probabilities a reasonable approach?

 

[FactChecker]

Does it means that the system 'typically' takes 0 time to transit to B in this path?

In the graph you have shown, even though the highest probability is at time 0 it is still a small fraction if the whole. There are still all the rest to consider. You should distinguish between "most probable" and "expected". The expected time is not the same as the peak time. The expected time is the average of all the results.

 

[FactChecker]

Does it means that the system 'typically' takes 0 time to transit to B in this path?

In the graph you have shown, even though the highest probability is at time 0 it is still a small fraction if the whole. There are still all the rest to consider. You should distinguish between "most probable" and "expected". The expected time is not the same as the time with highest probability. The expected time is the average of all the results, including those all the way out on the right hand tail of the distribution.

 

[phyalan]

In the graph you have shown, even though the highest probability is at time 0 it is still a small fraction if the whole. There are still all the rest to consider. You should distinguish between "most probable" and "expected". The expected time is not the same as the time with highest probability. The expected time is the average of all the results, including those all the way out on the right hand tail of the distribution.

Yes, I know that. But what I mean is in this case, how can one interpret the most probable first passage time where you have a peak at time 0? The point is sometimes, in some systems, the distribution has to very long tail that makes the mean fpt carries less significant meaning in describing the kinetics. So I what to know how to make sense our of this case.

 

[blue_raver22]

I would interpret the first passage time as the amount of time it takes for the system to transition from state A to state B for the first time. This can be thought of as the time it takes for the system to reach a certain threshold or criteria for transitioning to state B.

The non-zero peak at t=0 in the distribution can be interpreted as the probability of an immediate transition from state A to B without any intermediate stops. This means that there is a chance for the system to transition to state B as soon as it reaches state A.

In terms of estimating the most probable first passage time, taking the weighted mean of the two peaks with respect to their probabilities can be a reasonable approach. However, it is important to also consider other factors such as the variability in the system and any potential errors in the model. It may be beneficial to also consider other statistical methods for estimating the most probable first passage time, such as maximum likelihood estimation or Bayesian inference.