The objective is to investigate two emerging information technologies in graduate studies and scientific cooperation. Internet is the first technology. The open source is the second. They help each other in many ways. We investigate the joint influence of both. Results of complexity theory show the limitations of exact analysis. That explains popularity of heuristic algorithms. It is well known that efficiency of heuristics depends on the parameters. Thus we need some automatic procedures for tuning the heuristics. That helps comparing results of different heuristics. This enhance their efficiency, too. An initial presentation of the basic ideas is in (Mockus, 2000). Preliminary results of distance graduate studies are in (Mockus, 2006a). Examples of optimization of sequential statistical decisions are in (Mockus, 2006b). In this paper the theory and applications of Bayesian Heuristic Approach are discussed. In the next paper examples of Bayesian Approach to automated tuning of heurist...