  An $n$-dimensional source with memory is observed by $K$ isolated encoders
via parallel channels, who compress their observations to transmit to the
decoder via noiseless rate-constrained links while leveraging their memory of
the past. At each time instant, the decoder receives $K$ new codewords from the
observers, combines them with the past received codewords, and produces a
minimum-distortion estimate of the latest block of $n$ source symbols. This
scenario extends the classical one-shot CEO problem to multiple rounds of
communication with communicators maintaining the memory of the past.
  We extend the Berger-Tung inner and outer bounds to the scenario with
inter-block memory, showing that the minimum asymptotically (as $n \to \infty$)
achievable sum rate required to achieve a target distortion is bounded by
minimal directed mutual information problems. For the Gauss-Markov source
observed via $K$ parallel AWGN channels, we show that the inner bound is tight
and solve the corresponding minimal directed mutual information problem,
thereby establishing the minimum asymptotically achievable sum rate. Finally,
we explicitly bound the rate loss due to a lack of communication among the
observers; that bound is attained with equality in the case of identical
observation channels.
  The general coding theorem is proved via a new nonasymptotic bound that uses
stochastic likelihood coders and whose asymptotic analysis yields an extension
of the Berger-Tung inner bound to the causal setting. The analysis of the
Gaussian case is facilitated by reversing the channels of the observers.
