  Standard Bayesian inference can build models that combine information from
various sources, but this inference may not be reliable if components of a
model are misspecified. Cut inference, as a particular type of modularized
Bayesian inference, is an alternative which splits a model into modules and
cuts the feedback from the suspect module. Previous studies have focused on a
two-module case, but a more general definition of a ``module'' remains unclear.
We present a formal definition of a ``module'' and discuss its properties. We
formulate methods for identifying modules; determining the order of modules;
and building the cut distribution that should be used for cut inference within
an arbitrary directed acyclic graph structure. We justify the cut distribution
by showing that it not only cuts the feedback but also is the best
approximation satisfying this condition to the joint distribution in the
Kullback-Leibler divergence. We also extend cut inference for the two-module
case to a general multiple-module case via a sequential splitting technique and
demonstrate this via illustrative applications.
