  We propose two new measures for extracting the unique information in $X$ and
not $Y$ about a message $M$, when $X, Y$ and $M$ are joint random variables
with a given joint distribution. We take a Markov based approach, motivated by
questions in fair machine learning, and inspired by similar Markov-based
optimization problems that have been used in the Information Bottleneck and
Common Information frameworks. We obtain a complete characterization of our
definitions in the Gaussian case (namely, when $X, Y$ and $M$ are jointly
Gaussian), under the assumption of Gaussian optimality. We also examine the
consistency of our definitions with the partial information decomposition (PID)
framework, and show that these Markov based definitions achieve non-negativity,
but not symmetry, within the PID framework.
