Asymptotically Sufficient Partitions and Quantizations

We consider quantizations of observations represented by finite partitions of observation spaces. Partitions usually decrease the sensitivity of observations to their probability distributions. A sequence of quantizations is considered to be asymptotically sufficient for a statistical problem if the loss of sensitivity is asymptotically negligible. The sensitivity is measured by f-divergences of distributions or the closely related f-informations including the classical Shannon information. It is demonstrated that in some cases the maximization of f-divergences means the same as minimization of distortion of observations in the classical sense considered in mathematical statistics and information theory. The main result of the correspondence is a general sufficient condition for the asymptotic sufficiency of quantizations. Selected applications of this condition are studied leading to new simple criteria of asymptotic optimality for quantizations of vector-valued observations and observations on general Poisson processes