Interpolating between sampling and variational inference with infinite stochastic mixtures

Sampling and Variational Inference (VI) are two large families of methods for approximate inference with complementary strengths. Sampling methods excel at approximating arbitrary probability distributions, but can be inefficient. VI methods are efficient, but can fail when probability distributions are complex. Here, we develop a framework for constructing intermediate algorithms that balance the strengths of both sampling and VI. Both approximate a probability distribution using a mixture of simple component distributions: in sampling, each component is a delta-function and is chosen stochastically, while in standard VI a single component is chosen to minimize divergence. We show that sampling and VI emerge as special cases of an optimization problem over a mixing distribution, and intermediate approximations arise by varying a single parameter. We then derive closed-form sampling dynamics over variational parameters that stochastically build a mixture. Finally, we discuss how to select the optimal compromise between sampling and VI given a computational budget. This work is a first step towards a highly flexible yet simple family of inference methods that combines the complementary strengths of sampling and VI.