  Quantum Graphical Models (QGMs) generalize classical graphical models by
adopting the formalism for reasoning about uncertainty from quantum mechanics.
Unlike classical graphical models, QGMs represent uncertainty with density
matrices in complex Hilbert spaces. Hilbert space embeddings (HSEs) also
generalize Bayesian inference in Hilbert spaces. We investigate the link
between QGMs and HSEs and show that the sum rule and Bayes rule for QGMs are
equivalent to the kernel sum rule in HSEs and a special case of Nadaraya-Watson
kernel regression, respectively. We show that these operations can be
kernelized, and use these insights to propose a Hilbert Space Embedding of
Hidden Quantum Markov Models (HSE-HQMM) to model dynamics. We present
experimental results showing that HSE-HQMMs are competitive with
state-of-the-art models like LSTMs and PSRNNs on several datasets, while also
providing a nonparametric method for maintaining a probability distribution
over continuous-valued features.
