  Industrial applications frequently pose a notorious challenge for
state-of-the-art methods in the contexts of optimization, designing experiments
and modeling unknown physical response. This problem is aggravated by limited
availability of clean data, uncertainty in available physics-based models and
additional logistic and computational expense associated with experiments. In
such a scenario, Bayesian methods have played an impactful role in alleviating
the aforementioned obstacles by quantifying uncertainty of different types
under limited resources. These methods, usually deployed as a framework, allows
decision makers to make informed choices under uncertainty while being able to
incorporate information on the the fly, usually in the form of data, from
multiple sources while being consistent with the physical intuition about the
problem. This is a major advantage that Bayesian methods bring to fruition
especially in the industrial context. This paper is a compendium of the
Bayesian modeling methodology that is being consistently developed at GE
Research. The methodology, called GE's Bayesian Hybrid Modeling (GEBHM), is a
probabilistic modeling method, based on the Kennedy and O'Hagan framework, that
has been continuously scaled-up and industrialized over several years. In this
work, we explain the various advancements in GEBHM's methods and demonstrate
their impact on several challenging industrial problems.
