Classical approaches for approximate inference depend on cleverly designed variational distributions and bounds. Modern approaches employ amortized variational inference, which uses a neural network to approximate any posterior without leveraging the structures of the generative models. In this paper, we propose Amortized Conjugate Posterior (ACP), a hybrid approach taking advantages of both types of approaches. Specifically, we use the classical methods to derive specific forms of posterior distributions and then learn the variational parameters using amortized inference. We study the effectiveness of the proposed approach on the Noisy-OR model and compare to both the classical and the modern approaches for approximate inference and parameter learning. Our results show that ACP outperforms other methods when there is a limited amount of training data.

Estimating and optimizing Mutual Information (MI) is core to many problems in machine learning; however, bounding MI in high dimensions is challenging. To establish tractable and scalable objectives, recent work has turned to variational bounds parameterized by neural networks, but the relationships and tradeoffs between these bounds remains unclear. In this work, we unify these recent developments in a single framework. We find that the existing variational lower bounds degrade when the MI is large, exhibiting either high bias or high variance. To address this problem, we introduce a continuum of lower bounds that encompasses previous bounds and flexibly trades off bias and variance. On high-dimensional, controlled problems, we empirically characterize the bias and variance of the bounds and their gradients and demonstrate the effectiveness of our new bounds for estimation and representation learning.