Information-Theoretic Lower Bounds to an Average Distortion for the Models of Coding Independent Continuous Letters and Estimating a Distribution Parameter

A stochastic model of coding independent continuous letters distorted by a transmission channel and a similar model of estimating a random parameter of a given probability distribution density over independent samples have been investigated. For the models with a given distortion measure, the minimal values of an average mutual information between the processed data and all possible decisions depending on the fixed values of the average distortion are introduced. These trade-off relations are similar to the rate-distortion function that is well-known in the information theory. The lower bounds to the above relations have been constructed in the forms of strictly decreasing functions of the average mutual information by increasing the average distortion. The inversions of these bounds yield the appropriate lower bounds to the average distortion for the given values of the average mutual information. Independence of the obtained bounds on the decision algorithms permits to estimate an efficiency of any algorithm in terms of a redundancy of its information-theoretic characteristics relative to the boundary values.