  The recently introduced Gaussian Process State (GPS) provides a highly
flexible, compact and physically insightful representation of quantum many-body
states based on ideas from the zoo of machine learning approaches. In this
work, we give a comprehensive description how such a state can be learned from
given samples of a potentially unknown target state and show how regression
approaches based on Bayesian inference can be used to compress a target state
into a highly compact and accurate GPS representation. By application of a type
II maximum likelihood method based on Relevance Vector Machines (RVM), we are
able to extract many-body configurations from the underlying Hilbert space
which are particularly relevant for the description of the target state, as
support points to define the GPS. Together with an introduced optimization
scheme for the hyperparameters of the model characterizing the weighting of
modelled correlation features, this makes it possible to easily extract
physical characteristics of the state such as the relative importance of
particular correlation properties. We apply the Bayesian learning scheme to the
problem of modelling ground states of small Fermi-Hubbard chains and show that
the found solutions represent a systematically improvable trade-off between
sparsity and accuracy of the model. Moreover, we show how the learned
hyperparameters and the extracted relevant configurations, characterizing the
correlation of the wavefunction, depend on the interaction strength of the
Hubbard model as well as the target accuracy of the representation.
