  Modeling joint probability distributions over sequences has been studied from
many perspectives. The physics community developed matrix product states, a
tensor-train decomposition for probabilistic modeling, motivated by the need to
tractably model many-body systems. But similar models have also been studied in
the stochastic processes and weighted automata literature, with little work on
how these bodies of work relate to each other. We address this gap by showing
how stationary or uniform versions of popular quantum tensor network models
have equivalent representations in the stochastic processes and weighted
automata literature, in the limit of infinitely long sequences. We demonstrate
several equivalence results between models used in these three communities: (i)
uniform variants of matrix product states, Born machines and locally purified
states from the quantum tensor networks literature, (ii) predictive state
representations, hidden Markov models, norm-observable operator models and
hidden quantum Markov models from the stochastic process literature,and (iii)
stochastic weighted automata, probabilistic automata and quadratic automata
from the formal languages literature. Such connections may open the door for
results and methods developed in one area to be applied in another.
