Estimation of Dependences Based on Empirical Data

Twenty-five years have passed since the publication of the Russian version of the book Estimation of Dependencies Based on Empirical Data (EDBED for short). Twentyfive years is a long period of time. During these years many things have happened. Looking back, one can see how rapidly life and technology have changed, and how slow and difficult it is to change the theoretical foundation of the technology and its philosophy. I pursued two goals writing this Afterword: to update the technical results presented in EDBED (the easy goal) and to describe a general picture of how the new ideas developed over these years (a much more difficult goal). The picture which I would like to present is a very personal (and therefore very biased) account of the development of one particular branch of science, Empirical Inference Science. Such accounts usually are not included in the content of technical publications. I have followed this rule in all of my previous books. But this time I would like to violate it for the following reasons. First of all, for me EDBED is the important milestone in the development of empirical inference theory and I would like to explain why. Second, during these years, there were a lot of discussions between supporters of the new paradigm (now it is called the VC theory1) and the old one (classical statistics). Being involved in these discussions from the very beginning I feel that it is my obligation to describe the main events. The story related to the book, which I would like to tell, is the story of how it is difficult to overcome existing prejudices (both scientific and social), and how one should be careful when evaluating and interpreting new technical concepts. This story can be split into three parts that reflect three main ideas in the development of empirical inference science: from the pure technical (mathematical) elements of the theory to a new paradigm in the philosophy of generalization. The first part of the story, which describes the mai