  We develop an efficient algorithm to find optimal observation times by
maximizing the Fisher information for the birth rate of a partially observable
pure birth process involving $n$ observations. Partially observable implies
that at each of the $n$ observation time points for counting the number of
individuals present in the pure birth process, each individual is observed
independently with a fixed probability $p$, modeling detection difficulties or
constraints on resources. We apply concepts and techniques from generating
functions, using a combination of symbolic and numeric computation, to
establish a recursion for evaluating and optimizing the Fisher information. Our
numerical results reveal the efficacy of this new method. An implementation of
the algorithm is available publicly.
