  Variational inference is an umbrella term for algorithms which cast Bayesian
inference as optimization. Classically, variational inference uses the
Kullback-Leibler divergence to define the optimization. Though this divergence
has been widely used, the resultant posterior approximation can suffer from
undesirable statistical properties. To address this, we reexamine variational
inference from its roots as an optimization problem. We use operators, or
functions of functions, to design variational objectives. As one example, we
design a variational objective with a Langevin-Stein operator. We develop a
black box algorithm, operator variational inference (OPVI), for optimizing any
operator objective. Importantly, operators enable us to make explicit the
statistical and computational tradeoffs for variational inference. We can
characterize different properties of variational objectives, such as objectives
that admit data subsampling---allowing inference to scale to massive data---as
well as objectives that admit variational programs---a rich class of posterior
approximations that does not require a tractable density. We illustrate the
benefits of OPVI on a mixture model and a generative model of images.
