  Capturing the correlation emerging between constituents of many-body systems
accurately is one of the key challenges for the appropriate description of
various systems whose properties are underpinned by quantum mechanical
fundamentals. This thesis discusses novel tools and techniques for the
(classical) modelling of quantum many-body wavefunctions with the ultimate goal
to introduce a universal framework for finding accurate representations from
which system properties can be extracted efficiently. It is outlined how
synergies with standard machine learning approaches can be exploited to enable
an automated inference of the most relevant intrinsic characteristics through
rigorous Bayesian regression techniques. Based on the probabilistic framework
forming the foundation of the introduced ansatz, coined the Gaussian Process
State, different compression techniques are explored to extract numerically
feasible representations of relevant target states within stochastic schemes.
By following intuitively motivated design principles, the resulting model
carries a high degree of interpretability and offers an easily applicable tool
for the numerical study of quantum systems, including ones which are
notoriously difficult to simulate due to a strong intrinsic correlation. The
practical applicability of the Gaussian Process States framework is
demonstrated within several benchmark applications, in particular, ground state
approximations for prototypical quantum lattice models, Fermi-Hubbard models
and $J_1-J_2$ models, as well as simple ab-initio quantum chemical systems.
