This book attempts to lay down some minimal set of coherent and consistent requirements for Advanced Information Processors (AIPs). The idea is advanced that AIPs should reason in an optimal manner by generalizing logic. Logic is generalized through the auspices of probability theory. Probability is presented through the viewpoint of the formal manipulation rules such as Bayes's Theorem and the maximum entropy principle. Information Geometry is an alternative way to think about the mathematical characterization of entropy and the assignment of numerical values to probabilities. The approach emphasizes many solved numerical examples and relies upon the computational ability of Mathematica.