I've got a physical system that detect counts in an array of detectors. In each detector $y_i$ I expect to measure $\bar y_i = f_i(\bar \lambda)+\bar b_i$ counts. $b$ represent the vector of of counts that I expect to measure in the dector $i$ due to background noise, while $f_i(\lambda)$ is the number of counts that I expect to measure in detector $i$ due to the process of interest. Everything's Poisson.
$\lambda$ is the vector of parameters that I want to estimate. If the expectation value of $b$ is known, one can compute $\lambda$ from the measured counts $y$ maximizing the likelihood. It has also been shown that $b$ cannot be estimated from $y$ alone, it has to be measured independently. Luckily the electronics of the machine is able to do that quite well, therefore we can estimate $\lambda$ from $y$, giving in input to our ML algorithm our best estimate of $\bar b$ which we'll call $b'$. Keep in mind that what we are able to estimate (pretty well actually) is the expectaction value of $b$, not the number of actually collected counts due to $b$. But this is not a problem when modelling everything using the likelihood.
So, my problem is: What happens if my estimate $b'$ is wrong? How will my estimated $\lambda$ be influenced? I'd like to estimate this analytically if possible. How do I start addressing this? Shall I compute something like the expectation value of $\frac{\partial^2 L}{\partial \lambda_j \partial b_i}\Delta \lambda_j \Delta b_i = 0 $ at the likelihood maximum? Shall I just impose that $f_i(\lambda)+b_i = f_i(\lambda')+b'_i$ (i.e.: imposing that the expected number of counts is equal)?
