  A guiding principle for data reduction in statistical inference is the
sufficiency principle. This paper extends the classical sufficiency principle
to decentralized inference, i.e., data reduction needs to be achieved in a
decentralized manner. We examine the notions of local and global sufficient
statistics and the relationship between the two for decentralized inference
under different observation models. We then consider the impacts of
quantization on decentralized data reduction which is often needed when
communications among sensors are subject to finite capacity constraints. 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
optimal 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 under which sufficiency based data reduction
followed by quantization 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 observations that admit
conditional independence structure through the introduction of an appropriately
chosen hidden variable.
