  The posterior matching scheme, for feedback encoding of a message point lying
on the unit interval over memoryless channels, maximizes mutual information for
an arbitrary number of channel uses. However, it in general does not always
achieve any positive rate; so far, elaborate analyses have been required to
show that it achieves any positive rate below capacity. More recent efforts
have introduced a random "dither" shared by the encoder and decoder to the
problem formulation, to simplify analyses and guarantee that the randomized
scheme achieves any rate below capacity. Motivated by applications (e.g.
human-computer interfaces) where (a) common randomness shared by the encoder
and decoder may not be feasible and (b) the message point lies in a higher
dimensional space, we focus here on the original formulation without common
randomness, and use optimal transport theory to generalize the scheme for a
message point in a higher dimensional space. By defining a stricter, almost
sure, notion of message decoding, we use classical probabilistic techniques
(e.g. change of measure and martingale convergence) to establish succinct
necessary and sufficient conditions on when the message point can be recovered
from infinite observations: Birkhoff ergodicity of a random process
sequentially generated by the encoder. We also show a surprising "all or
nothing" result: the same ergodicity condition is necessary and sufficient to
achieve any rate below capacity. We provide applications of this message point
framework in human-computer interfaces and multi-antenna communications.
