Lower bound to estimation distortion of a random parameter for a given amount of information

Given probability distribution density with an unknown value of a random parameter, a minimum of the average square distortion for the parameter estimates via the samples of random values as a function of the average mutual information between the samples and the estimates is investigated. This function is produced by inverting a modified rate distortion function as the dependency of the minimal values of the average mutual information on the appropriate values of the average distortion. The obtained smallest average square distortion as the function of the average mutual information is independent on an estimation form and this function yields the lower bound to the average distortion for the fixed values of the amount of information. The above relation is the bifactor fidelity decision criterion that allows one to compare various estimation functions by their efficiency in terms of the average distortion redundancy relative to the lower bound when the entropy of the quantized estimates is fixed.