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Hello! I have a setup consisting of some charged particles each of which is produced at a different position, ##(x_i,y_i,z_i)##. I don't know the exact position, but I know that each of the 3 variables is normally distributed with mean zero and standard deviation of 3 mm. What I measure in the experiment is the position of the particles on a 2D screen perpendicular to the trajectory of motion of the particles, call it x' and y' and the time from the moment they are produced to when they hit the 2D screen, call it t. So what i measure is ##(x',y',t)## with the associated uncertainties, given by the detector, ##(dx',dy',dt)##. What I want to do is, for each ion, from these measurements, to get the most probable starting point (together with the associated uncertainty). I am not sure what is the best way to proceed. What I want to get basically is:

$$p(x_i,y_i,z_i|x_f,y_f,t)$$

and I will write this just as ##p(x_i|x_f)## for simplicity. For example, assuming we had no uncertainty on our measurements, I could use Bayes theorem and have:

$$p(x_i|x_f) = \frac{p(x_f|x_i)p(x_i)}{p(x_f)}$$

I know ##p(x_i)## (which is Gaussian) and for a given event, ##p(x_f)## is constant (I don't know what it is, but it is constant). For ##p(x_f|x_i)## I can get a value by running some simulations. Assuming everything would be ideal, this would be 1 for just one value and 0 for everything else. Given that we can't model the electrodes perfectly, and I will run the simulation several times with slightly different potentials, this will probably have a distribution, too (say Gaussian for simplicity). From here I can get ##p(x_i|x_f)## and the associated standard deviation, which is exactly what I need.

However, given that I have noise on ##(x_f,y_f,t)##, I can't just apply the Bayes theorem directly as above. How can I account for these errors? Thank you!

EDIT: I attached before a diagram of the trajectory of the particle (red line) from the source to the 2D detector.

$$p(x_i,y_i,z_i|x_f,y_f,t)$$

and I will write this just as ##p(x_i|x_f)## for simplicity. For example, assuming we had no uncertainty on our measurements, I could use Bayes theorem and have:

$$p(x_i|x_f) = \frac{p(x_f|x_i)p(x_i)}{p(x_f)}$$

I know ##p(x_i)## (which is Gaussian) and for a given event, ##p(x_f)## is constant (I don't know what it is, but it is constant). For ##p(x_f|x_i)## I can get a value by running some simulations. Assuming everything would be ideal, this would be 1 for just one value and 0 for everything else. Given that we can't model the electrodes perfectly, and I will run the simulation several times with slightly different potentials, this will probably have a distribution, too (say Gaussian for simplicity). From here I can get ##p(x_i|x_f)## and the associated standard deviation, which is exactly what I need.

However, given that I have noise on ##(x_f,y_f,t)##, I can't just apply the Bayes theorem directly as above. How can I account for these errors? Thank you!

EDIT: I attached before a diagram of the trajectory of the particle (red line) from the source to the 2D detector.

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