Divide and Couple: Using Monte Carlo Variational Objectives for Posterior Approximation
This work addresses a foundational gap in variational inference for researchers and practitioners needing accurate posterior approximations, though it appears incremental as it builds on existing Monte Carlo methods.
The paper tackles the problem of connecting variational inference objectives to posterior approximation by introducing a 'divide and couple' procedure that identifies augmented distributions, enabling the use of Monte Carlo variational objectives for approximating the posterior distribution after optimization.
Recent work in variational inference (VI) uses ideas from Monte Carlo estimation to tighten the lower bounds on the log-likelihood that are used as objectives. However, there is no systematic understanding of how optimizing different objectives relates to approximating the posterior distribution. Developing such a connection is important if the ideas are to be applied to inference-i.e., applications that require an approximate posterior and not just an approximation of the log-likelihood. Given a VI objective defined by a Monte Carlo estimator of the likelihood, we use a "divide and couple" procedure to identify augmented proposal and target distributions. The divergence between these is equal to the gap between the VI objective and the log-likelihood. Thus, after maximizing the VI objective, the augmented variational distribution may be used to approximate the posterior distribution.