  We investigate a correspondence between two formalisms for discrete
probabilistic modeling: probabilistic graphical models (PGMs) and tensor
networks (TNs), a powerful modeling framework for simulating complex quantum
systems. The graphical calculus of PGMs and TNs exhibits many similarities,
with discrete undirected graphical models (UGMs) being a special case of TNs.
However, more general probabilistic TN models such as Born machines (BMs)
employ complex-valued hidden states to produce novel forms of correlation among
the probabilities. While representing a new modeling resource for capturing
structure in discrete probability distributions, this behavior also renders the
direct application of standard PGM tools impossible. We aim to bridge this gap
by introducing a hybrid PGM-TN formalism that integrates quantum-like
correlations into PGM models in a principled manner, using the
physically-motivated concept of decoherence. We first prove that applying
decoherence to the entirety of a BM model converts it into a discrete UGM, and
conversely, that any subgraph of a discrete UGM can be represented as a
decohered BM. This method allows a broad family of probabilistic TN models to
be encoded as partially decohered BMs, a fact we leverage to combine the
representational strengths of both model families. We experimentally verify the
performance of such hybrid models in a sequential modeling task, and identify
promising uses of our method within the context of existing applications of
graphical models.
