Recent advances in statistical inference have significantly expanded the toolbox of probabilistic modeling. Historically, probabilistic modeling has been constrained to very restricted model classes, where exact or approximate probabilistic inference is feasible. However, developments in variational inference, a general form of approximate probabilistic inference that originated in statistical physics, have enabled probabilistic modeling to overcome these limitations: (i) Approximate probabilistic inference is now possible over a broad class of probabilistic models containing a large number of parameters, and (ii) scalable inference methods based on stochastic gradient descent and distributed computing engines allow probabilistic modeling to be applied to massive data sets. One important practical consequence of these advances is the possibility to include deep neural networks within probabilistic models, thereby capturing complex non-linear stochastic relationships between the random variables. These advances, in conjunction with the release of novel probabilistic modeling toolboxes, have greatly expanded the scope of applications of probabilistic models, and allowed the models to take advantage of the recent strides made by the deep learning community. In this paper, we provide an overview of the main concepts, methods, and tools needed to use deep neural networks within a probabilistic modeling framework.

In many modern data analysis problems, the available data is not static but, instead, comes in a streaming fashion. Performing Bayesian inference on a data stream is challenging for several reasons. First, it requires continuous model updating and the ability to handle a posterior distribution conditioned on an unbounded data set. Secondly, the underlying data distribution may drift from one time step to another, and the classic i.i.d. (independent and identically distributed), or data exchangeability assumption does not hold anymore. In this paper, we present an approximate Bayesian inference approach using variational methods that addresses these issues for conjugate exponential family models with latent variables. Our proposal makes use of a novel scheme based on hierarchical priors to explicitly model temporal changes of the model parameters. We show how this approach induces an exponential forgetting mechanism with adaptive forgetting rates. The method is able to capture the smoothness of the concept drift, ranging from no drift to abrupt drift. The proposed variational inference scheme maintains the computational efficiency of variational methods over conjugate models, which is critical in streaming settings. The approach is validated on four different domains (energy, finance, geolocation, and text) using four real-world data sets.