Are global sufficient statistics always sufficient: The impact of quantization on decentralized data reduction

The sufficiency principle is the guiding principle for data reduction for various statistical inference problems. There has been recent effort in developing the sufficiency principle for decentralized inference with a particular emphasis on studying the relationship between global sufficient statistics and local sufficient statistics. We consider in this paper the impact of quantization on decentralized data reduction. The central question we intend to ask is: if each node in a decentralized inference system has to summarize its data using a finite number of bits, is it still sufficient to implement data reduction using global sufficient statistics prior to quantization? We show that the answer is negative using a simple example and proceed to identify conditions when global sufficient statistics based data reduction is indeed optimal. They include the well known case when the data at decentralized nodes are conditionally independent as well as a class of problems with conditionally dependent data.