The book presents a basic theory of statistical inference and information with emphasis on methods of convex analysis common for mathematical statistics and information theory. The level of mathematics used is based on abstract measurable spaces wherefore the book is best suited for postgraduate and advanced undergraduate courses on the topic. The book studies systematically convex functions of measures in general, and convex functions of probability measures in particular. Distances of probability measures used in mathematical statistics and information theory are examples of these convex functions. Applications of these functions in mathematical statistics and information theory (e.g. concepts of information, Bayes risk, entropy, Kantorovich-Vasershtein and Ornstein distances, etc.) are in a central role in the book. Common tools used in and relations existing between mathematical statistics and information theory are considered systematically.
Reviewer: J.Virtanen