Symbolic analysis as universal tool for deriving properties of non-linear algorithms - Case study of EM algorithm

Many researchers start their work by studying theory in order to get better insight into measured phenomena. Sometimes they cannot get the numeric values of parameters used in published equations. This is even more complicated when the theory is statistical and there are no closed form deterministic solutions. In this paper we introduce an original approach and method to analyzing a popular and frequently cited tutorial paper on Expectation-Maximization (EM) algorithm. The original paper has sufficient information to understand the algorithm. Using tools of Computer Algebra System and methods of Symbolic Processing (SP), the typewriting errors are discovered and the re-derived equations are used for automatic generation of algorithmic code. The examples are evaluated using automatically derived code and the final numeric values agree with the values from the original paper. The derived results are used for further optimization, such as deriving the computational error in the closed form. From the closed form solutions, the precision can be derived in terms of number of iterations, or the minimal number of iterations can be expressed in terms of the required precision. This helps to optimize the algorithm parameters so that the algorithm becomes more efficient.