Rate distortion function with causal decoding

This paper considers source coding of general sources with memory, when causal feedback is available at the decoder. The rate distortion function is defined as the infimum of the so-called directed information over causal data compression channels which satisfy a distortion fidelity constraint. The expression of the optimal causal data compression is derived. Further a tight lower bound on the rate distortion bound is derived. These are variants of the classical non-causal rate distortion function, and associate Shannon lower bound. Generalization of the results to controlled sources is also discussed.