  In a variety of scientific applications we wish to characterize a physical
system using measurements or observations. This often requires us to solve an
inverse problem, which usually has non-unique solutions so uncertainty must be
quantified in order to define the family of all possible solutions. Bayesian
inference provides a powerful theoretical framework which defines the set of
solutions to inverse problems, and variational inference is a method to solve
Bayesian inference problems using optimization while still producing fully
probabilistic solutions. This chapter provides an introduction to variational
inference, and reviews its applications to a range of geophysical problems,
including petrophysical inversion, travel time tomography and full-waveform
inversion. We demonstrate that variational inference is an efficient and
scalable method which can be deployed in many practical scenarios.
