While it's always possible to compute a variational approximation to a posterior distribution, it can be difficult to discover problems with this approximation". We propose two diagnostic algorithms to alleviate this problem. The Pareto-smoothed importance sampling (PSIS) diagnostic gives a goodness of fit measurement for joint distributions, while simultaneously improving the error in the estimate. The variational simulation-based calibration (VSBC) assesses the average performance of point estimates.

Categorical distributions are ubiquitous in machine learning, e.g., in classification, language models, and recommendation systems. However, when the number of possible outcomes is very large, using categorical distributions becomes computationally expensive, as the complexity scales linearly with the number of outcomes. To address this problem, we propose augment and reduce (A&R), a method to alleviate the computational complexity. A&R uses two ideas: latent variable augmentation and stochastic variational inference. It maximizes a lower bound on the marginal likelihood of the data. Unlike existing methods which are specific to softmax, A&R is more general and is amenable to other categorical models, such as multinomial probit. On several large-scale classification problems, we show that A&R provides a tighter bound on the marginal likelihood and has better predictive performance than existing approaches.

Parameter inference for stochastic differential equations is challenging due to the presence of a latent diffusion process. Working with an Euler-Maruyama discretisation for the diffusion, we use variational inference to jointly learn the parameters and the diffusion paths. We use a standard mean-field variational approximation of the parameter posterior, and introduce a recurrent neural network to approximate the posterior for the diffusion paths conditional on the parameters. This neural network learns how to provide Gaussian state transitions which bridge between observations in a very similar way to the conditioned diffusion process. The resulting black-box inference method can be applied to any SDE system with light tuning requirements. We illustrate the method on a Lotka-Volterra system and an epidemic model, producing accurate parameter estimates in a few hours.

Amortized inference allows latent-variable models trained via variational learning to scale to large datasets. The quality of approximate inference is determined by two factors: a) the capacity of the variational distribution to match the true posterior and b) the ability of the recognition network to produce good variational parameters for each datapoint. We examine approximate inference in variational autoencoders in terms of these factors. We find that divergence from the true posterior is often due to imperfect recognition networks, rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.