We develop a general variational inference method that preserves dependency among the latent variables. Our method uses copulas to augment the families of distributions used in mean-field and structured approximations. Copulas model the dependency that is not captured by the original variational distribution, and thus the augmented variational family guarantees better approximations to the posterior. With stochastic optimization, inference on the augmented distribution is scalable. Furthermore, our strategy is generic: it can be applied to any inference procedure that currently uses the mean-field or structured approach. Copula variational inference has many advantages: it reduces bias; it is less sensitive to local optima; it is less sensitive to hyperparameters; and it helps characterize and interpret the dependency among the latent variables.

We develop a general variational inference method that preserves dependency among the latent variables. Our method uses copulas to augment the families of distributions used in mean-field and structured approximations. Copulas model the dependency that is not captured by the original variational distribution, and thus the augmented variational family guarantees better approximations to the posterior. With stochastic optimization, inference on the augmented distribution is scalable. Furthermore, our strategy is generic: it can be applied to any inference procedure that currently uses the mean-field or structured approach. Copula variational inference has many advantages: it reduces bias; it is less sensitive to local optima; it is less sensitive to hyperparameters; and it helps characterize and interpret the dependency among the latent variables.

We develop methods for parameter estimation in settings with large-scale data sets, where traditional methods are no longer tenable. Our methods rely on stochastic approximations, which are computationally efficient as they maintain one iterate as a parameter estimate, and successively update that iterate based on a single data point. When the update is based on a noisy gradient, the stochastic approximation is known as standard stochastic gradient descent, which has been fundamental in modern applications with large data sets. Additionally, our methods are numerically stable because they employ implicit updates of the iterates. Intuitively, an implicit update is a shrinked version of a standard one, where the shrinkage factor depends on the observed Fisher information at the corresponding data point. This shrinkage prevents numerical divergence of the iterates, which can be caused either by excess noise or outliers. Our sgd package in R offers the most extensive and robust implementation of stochastic gradient descent methods. We demonstrate that sgd dominates alternative software in runtime for several estimation problems with massive data sets. Our applications include the wide class of generalized linear models as well as M-estimation for robust regression.

We develop a robust convex algorithm to select the regularization parameter in model selection. In practice this would be automated in order to save practitioners time from having to tune it manually. In particular, we implement and test the convex method for $K$-fold cross validation on ridge regression, although the same concept extends to more complex models. We then compare its performance with standard methods.