Statistical information and discrimination

In analogy with the definition of Shannon information, M.H. De Groot (1962) defined statistical information as the difference between prior and posterior risk of a statistical decision problem. Relations are studied between the statistical information and the discrimination functions of information theory known as f-divergences. Using previous results, it is shown that every f-divergence I_f(P,Q) is an average statistical information or decision problem with dichotomic parameter, 0-1 loss function, and corresponding observation distributions P and Q. The average is taken over a distribution on the parameter's prior probability. This distribution is uniquely determined by the function f. The main result is that every f-divergence is statistical information in some properly chosen statistical decision problem, and conversely, that every piece of statistical information is an f-divergence. This provides a new representation of discrimination functions figuring in signal detection, data compression, coding pattern classification, cluster analysis, etc