Some Topics in Statistical Information Theory
GEORGE WASHINGTON UNIV WASHINGTON DC DEPT OF STATISTICS
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Attention is focused on informational properties of sub-sigma-algebras of the fundamental probability space in contrast to the discussion in Information Theory and Statistics Wiley 1959, Dover Publications Inc. 1968 where attention is devoted to informational properties of statistics, that is, random variables. Properties of relative conditional expectations and results on separable sigma-algebras are stated for later use. The discrimination information in a sigma-algebra sub-sigma-algebra is defined and various properties developed, in particular the relation between information and sufficiency. The integral representative theorem for discrimination information is derived by methods believed to be more inherently information-theoretic than others that have been presented. Monotonic properties of conditional discrimination information are derived with respect to either the conditioning sub-sigma-algebra or the conditioned sub-sigma-algebra.