Statistical inference under multiterminal rate restrictions: a differential geometric approach

A statistical inference problem for a two-terminal information source emitting mutually correlated signals X and Y is treated. Let sequences X/sup n/ and Y/sup n/ of n independent observations be encoded independently of each other into message sets M/sub X/ and M/sub Y/ at rates R/sub 1/ and R/sub 2/ per letter, respectively. This compression causes a loss of the statistical information available for testing hypotheses concerning X and Y. The loss of statistical information is evaluated as a function of the amounts R/sub 1/ and R/sub 2/ of the Shannon information. A complete solution is given in the case of asymptotically complete data compression, R/sub 1/, R/sub 2/ to 0 as n to infinity . It is shown that the differential geometry of the manifold of all probability distributions plays a fundamental role in this type of multiterminal problem connecting Shannon information and statistical information. A brief introduction to the dually coupled e-affine and m-affine connections together with e-flatness and m-flatness is given.>