  We investigate distributed source coding of two correlated sources X and Y
where messages are passed to a decoder in a cascade fashion. The encoder of X
sends a message at rate R_1 to the encoder of Y. The encoder of Y then sends a
message to the decoder at rate R_2 based both on Y and on the message it
received about X. The decoder's task is to estimate a function of X and Y. For
example, we consider the minimum mean squared-error distortion when encoding
the sum of jointly Gaussian random variables under these constraints. We also
characterize the rates needed to reconstruct a function of X and Y losslessly.
  Our general contribution toward understanding the limits of the cascade
multiterminal source coding network is in the form of inner and outer bounds on
the achievable rate region for satisfying a distortion constraint for an
arbitrary distortion function d(x,y,z). The inner bound makes use of a balance
between two encoding tactics--relaying the information about X and
recompressing the information about X jointly with Y. In the Gaussian case, a
threshold is discovered for identifying which of the two extreme strategies
optimizes the inner bound. Relaying outperforms recompressing the sum at the
relay for some rate pairs if the variance of X is greater than the variance of
Y.
