A novel sequential RDF to compute partially observable Markov processes with MSE distortion

<div>We develop a new sequential rate distortion function</div><div>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.</div><div>Under the assumption that all matrices commute by pairs,</div><div>we show that our problem can be cast as a convex program</div><div>which achieves its global minimum. We also derive sufficient</div><div>conditions which ensure that our assumption holds. We then</div><div>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.</div>