I have a count-data of 3 different vehicle classes for a couple of streets. I know that all the count data of vehicles from streets are miscounted, in a sense that, some class A vehicles (bikes) counted as class B vehicles (Car), and some Cars counted as class C vehicles (Truck), However total count is correct.
I know that some streets are forbidden to motorized vehicles (class B and Class C Vehicle). Hence, I feel like those streets are a good source for modeling the 'misclassification rate' of our system.
I want to estimate the number of daily (I have hourly/daily data) counts for each class of vehicles given the data while considering the "misclassification rate".
I know that poison distribution is used for count distributions. Should I use a compound poison dist for modeling the total number of passing vehicles (since it is the only correct one)? However, I don't know how to integrate the misclassification in this case, my intuition says that the only way to add misclassification info as a prior in a Bayesian context.
To summarize, I don't know how to approach the problem, any model recommendations or theories would be more than welcome
You have a $3 \times 3$ matrix of probabilities. If you consider the numbers of vehicles a column vector of length $3$ for the three types the matrix will take the actual number of vehicles and transform it to the counted number of vehicles. You need to collect the data to fill out that matrix. You can then invert it to go from counted data to "actual" figures.
The usual way to do this would be to get some ground truth. Mount cameras over some of your roads and review the recordings to get the actual numbers. Compare that with the output of your counters.
It is not important to select a particular distribution. That is what you are measuring.