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We study data processing inequalities that are derived from a certain classof generalized information measures, where a series of convex functions andmultiplicative likelihood ratios are nested alternately. While theseinformation measures can be viewed as a special case of the most generalZakai-Ziv generalized information measure, this special nested structure callsfor attention and motivates our study. Specifically, a certain choice of theconvex functions leads to an information measure that extends the notion of theBhattacharyya distance (or the Chernoff divergence): While the ordinaryBhattacharyya distance is based on the (weighted) geometric mean of tworeplicas of the channel's conditional distribution, the more generalinformation measure allows an arbitrary number of such replicas. We apply thedata processing inequality induced by this information measure to a detailedstudy of lower bounds of parameter estimation under additive white Gaussiannoise (AWGN) and show that in certain cases, tighter bounds can be obtained byusing more than two replicas. While the resulting lower bound may not competefavorably with the best bounds available for the ordinary AWGN channel, theadvantage of the new lower bound, relative to the other bounds, becomessignificant in the presence of channel uncertainty, like unknown fading. Thisdifferent behavior in the presence of channel uncertainty is explained by theconvexity property of the information measure.

Data processing inequalities based on a certain structured class of information measures with application to estimation theory