The rate distortion function for a class of sources

For a source with a known probability distribution, α, Shannon's rate distortion function, Rα(d), specifies the minimum channel capacity required to transmit the output of the source with average distortion ≦ d. In practice, the encoding system must often be designed when only vague information is available concerning the source distribution; i.e., it is known only that the source distribution is a member of some class α. It is thus important to extend Shannon's theory to cover this case. In this paper we define a rate distortion function for a class of sources, Rα(d), and prove a coding theorem. This theorem establishes that Rα(d) is the minimum channel capacity required by any system which can transmit each source in α with average distortion ≦ d, and that for a compact class of sources this rate can be approached as closely as desired. Further, we show that for a compact class, Rα(d) is the sup of Rα(d) over all α in α. This last result is important in that it greatly simplifies the computation of Rα(d)