Implementation of Poisson experiment

There are three difficulties implementing an experiment of the Poisson distribution compared with the binomial distribution. Firstly, the problem as stated so far is incomplete. Poisson-distributed random variables normally produce a list of scalars, not a list of 2-tuples. If we consider the problem as producing a list of scalars c_i, then it is legitimate to ask the distribution of the scalars t_i. (The process can also be perform in reverse by assuming that t_i is a variable and c_i comes from some hidden distribution.) It does not affect the equations as the likelihood function has factored in that t_i is a variable. However, when implementing a numerical solution, we must know what the data ``looks'' like - otherwise, sample data cannot be generated. In this appendix, I've assumed that $t_i = 1~\forall~i$. That is, t_i has a constant distribution. Just as legitimately, t_i could have a uniform distribution, Poisson distribution or some other distribution.

The second problem is the domain of c_i. The binomial problem produces a number between 0 and N. Therefore, provided N is relatively low, it is a realistic solution to integrate (for any $\theta$) over the domain of m. Indeed, that was the technique used to evaluate the Expected Kullback-Leibler and Expected Root-Mean-Square distances. The Poisson distribution produces numbers (for any r) from zero to infinity. Obviously, integration over the whole domain is not a realistic solution. However, the probability for large values of c_i, is relatively small. Furthermore, the probability decreases at a super-exponential rate (factorial actually). As a result, integration is still a possibility, although care must be taken to ensure a cut-off point. This can be implemented by ignoring any data-point c_i higher than a certain number c_max(r, t_i). In this appendix, it is assumed that c_max(r, t_i) = c_max(r, 1) = 10r.

The third problem is the fundamental difference between the binomial problem posed and the Poisson problem. With the binomial problem, the data-point consisted of a single number. With the Poisson problem, a list of N numbers (between 0 and c_max(r, t_i)) are produced. In general, there are (c_max(r, t_i)+1)^N possible sequences to integrate over - which is simply unrealistic. Therefore, some form of Monte Carlo simulation is required. This introduces uncertainly as an adequate sample size must be chosen and the random number generator must of a sufficiently high quality.