We develop a new sequential rate distortion function
to compute lower bounds on the average length of all causal prefix free codes for partially observable multivariate Markov processes with mean-squared error distortion constraint. Our information measure is characterized by a variant of causally conditioned directed information and is utilized in various application examples. First, it is used to optimally characterize a finite dimensional optimization problem for jointly Gaussian processes and to obtain the corresponding optimal linear encoding and decoding policies.
Under the assumption that all matrices commute by pairs,
we show that our problem can be cast as a convex program
which achieves its global minimum. We also derive sufficient
conditions which ensure that our assumption holds. We then
solve the KKT conditions and derive a new reverse-waterfilling algorithm that we implement. If our assumption is violated, one can still use our approach to derive sub-optimal (upper bound) waterfilling solutions. For scalar-valued Gauss-Markov processes with additional observation noise, we derive a new closed form solution and we compare it with known results in the literature. For partially observable time-invariant Markov processes driven by additive i:i:d: system noise only, we recover using an alternative approach and thus strengthening a recent result by Kostina and Hassibi in [1, Theorem 9] whereas for timeinvariant and spatially IID Markov processes driven by additive noise process we also derive new analytical lower bounds.