I am trying to work on a nano scale communication problem. The transmission media is considered to be fluid and the molecules are the one that communicates information from transmitter to receiver. I am assuming that the receiver estimates the arrival rate(number of molecules per unit given time slot) and tells which hypothesis was there. We have hypothesis given as$:\\ H_0: \text {no molecule was transferred}\\ H_1: \text {one molecule was transferred}\\ H_2: \text {two molecules were transferred}\\ .\\ .\\ H_M:\text {M molecules were transferred}\\$
I have taken the arrival of molecules as a Poisson counting process and $\lambda$ is the observation variable which means that the receiver will calculate vale of arrival rate by dividing number of molecules received in the given time slot and make a judgement about correct hypothesis. After some calculation I get Probability of Error as a function of$:P_{Error}=f(\lambda,n,t)$ where I assume $\lambda$ to be a continuous variable, $n $ and $t$ are fixed.The receiver will decide a threshold for every hypothesis and will give decision about hypothesis $H_k$ if the calculated arrival rate lies between $\lambda_{k-\epsilon}$ and $\lambda_{k+\epsilon}$ where $\epsilon$ is around the arrival rate, and is the region in which if my observed rate falls I will say that it belongs to hypothesis $ k.\\ \\$
Now my problem is, the process is such that I don't know the arrival rate, have it been a binary hypothesis problem I would have fixed vale of $P_{F}$ and using Neyman Pearson approach will try to maximize $P_{D}$. But how should I do the same for M-ary Hypothesis. Further I am not getting the results in $P_F$ and $P_D$ terms but rather in $P_{Error}$ and $P_{NoError}$ terms. How should I proceed? Can Bayesian approach be applied in this? I do not know costs.
