Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient variational inference algorithm, freeing the scientist to refine and explore many models. ADVI supports a broad class of models-no conjugacy assumptions are required. We study ADVI across ten different models and apply it to a dataset with millions of observations. ADVI is integrated into Stan, a probabilistic programming system; it is available for immediate use.

Developing efficient and scalable algorithms for Latent Dirichlet Allocation (LDA) is of wide interest for many applications. Previous work has developed an O(1) Metropolis-Hastings sampling method for each token. However, the performance is far from being optimal due to random accesses to the parameter matrices and frequent cache misses. In this paper, we first carefully analyze the memory access efficiency of existing algorithms for LDA by the scope of random access, which is the size of the memory region in which random accesses fall, within a short period of time. We then develop WarpLDA, an LDA sampler which achieves both the best O(1) time complexity per token and the best O(K) scope of random access. Our empirical results in a wide range of testing conditions demonstrate that WarpLDA is consistently 5-15x faster than the state-of-the-art Metropolis-Hastings based LightLDA, and is comparable or faster than the sparsity aware F+LDA. With WarpLDA, users can learn up to one million topics from hundreds of millions of documents in a few hours, at an unprecedentedly throughput of 11G tokens per second.

Herding defines a deterministic dynamical system at the edge of chaos. It generates a sequence of model states and parameters by alternating parameter perturbations with state maximizations, where the sequence of states can be interpreted as "samples" from an associated MRF model. Herding differs from maximum likelihood estimation in that the sequence of parameters does not converge to a fixed point and differs from an MCMC posterior sampling approach in that the sequence of states is generated deterministically. Herding may be interpreted as a"perturb and map" method where the parameter perturbations are generated using a deterministic nonlinear dynamical system rather than randomly from a Gumbel distribution. This chapter studies the distinct statistical characteristics of the herding algorithm and shows that the fast convergence rate of the controlled moments may be attributed to edge of chaos dynamics. The herding algorithm can also be generalized to models with latent variables and to a discriminative learning setting. The perceptron cycling theorem ensures that the fast moment matching property is preserved in the more general framework.

In a multi-class classification problem, it is standard to model the output of a neural network as a categorical distribution conditioned on the inputs. The output must therefore be positive and sum to one, which is traditionally enforced by a softmax. This probabilistic mapping allows to use the maximum likelihood principle, which leads to the well-known log-softmax loss. However the choice of the softmax function seems somehow arbitrary as there are many other possible normalizing functions. It is thus unclear why the log-softmax loss would perform better than other loss alternatives. In particular Vincent et al. (2015) recently introduced a class of loss functions, called the spherical family, for which there exists an efficient algorithm to compute the updates of the output weights irrespective of the output size. In this paper, we explore several loss functions from this family as possible alternatives to the traditional log-softmax. In particular, we focus our investigation on spherical bounds of the log-softmax loss and on two spherical log-likelihood losses, namely the log-Spherical Softmax suggested by Vincent et al. (2015) and the log-Taylor Softmax that we introduce. Although these alternatives do not yield as good results as the log-softmax loss on two language modeling tasks, they surprisingly outperform it in our experiments on MNIST and CIFAR-10, suggesting that they might be relevant in a broad range of applications.

Hyper-parameters play a major role in the learning and inference process of latent Dirichlet allocation (LDA). In order to begin the LDA latent variables learning process, these hyper-parameters values need to be pre-determined. We propose an extension for LDA that we call 'Latent Dirichlet allocation Gibbs Newton' (LDA-GN), which places non-informative priors over these hyper-parameters and uses Gibbs sampling to learn appropriate values for them. At the heart of LDA-GN is our proposed 'Gibbs-Newton' algorithm, which is a new technique for learning the parameters of multivariate Polya distributions. We report Gibbs-Newton performance results compared with two prominent existing approaches to the latter task: Minka's fixed-point iteration method and the Moments method. We then evaluate LDA-GN in two ways: (i) by comparing it with standard LDA in terms of the ability of the resulting topic models to generalize to unseen documents; (ii) by comparing it with standard LDA in its performance on a binary classification task.

The plausibility of uncommon events and miracles based on testimony of such an event has been much discussed. When analyzing the probabilities involved, it has mostly been assumed that the common events can be taken as data in the calculations. However, we usually have only testimonies for the common events. While this difference does not have a significant effect on the inductive part of the inference, it has a large influence on how one should view the reliability of testimonies. In this work, a full Bayesian solution is given for the more realistic case, where one has a large number of testimonies for a common event and one testimony for an uncommon event. It is seen that, in order for there to be a large amount of testimonies for a common event, the testimonies will probably be quite reliable. For this reason, because the testimonies are quite reliable based on the testimonies for the common events, the probability for the uncommon event, given a testimony for it, is also higher. Hence, one should be more open-minded when considering the plausibility of uncommon events.

A family of parsimonious Gaussian cluster-weighted models is presented. This family concerns a multivariate extension to cluster-weighted modelling that can account for correlations between multivariate responses. Parsimony is attained by constraining parts of an eigen-decomposition imposed on the component covariance matrices. A sufficient condition for identifiability is provided and an expectation-maximization algorithm is presented for parameter estimation. Model performance is investigated on both synthetic and classical real data sets and compared with some popular approaches. Finally, accounting for linear dependencies in the presence of a linear regression structure is shown to offer better performance, vis-\`{a}-vis clustering, over existing methodologies.

We propose a Bayesian approximate inference method for learning the dependence structure of a Gaussian graphical model. Using pseudo-likelihood, we derive an analytical expression to approximate the marginal likelihood for an arbitrary graph structure without invoking any assumptions about decomposability. The majority of the existing methods for learning Gaussian graphical models are either restricted to decomposable graphs or require specification of a tuning parameter that may have a substantial impact on learned structures. By combining a simple sparsity inducing prior for the graph structures with a default reference prior for the model parameters, we obtain a fast and easily applicable scoring function that works well for even high-dimensional data. We demonstrate the favourable performance of our approach by large-scale comparisons against the leading methods for learning non-decomposable Gaussian graphical models. A theoretical justification for our method is provided by showing that it yields a consistent estimator of the graph structure.

Link prediction is a fundamental task in statistical network analysis. Recent advances have been made on learning flexible nonparametric Bayesian latent feature models for link prediction. In this paper, we present a max-margin learning method for such nonparametric latent feature relational models. Our approach attempts to unite the ideas of max-margin learning and Bayesian nonparametrics to discover discriminative latent features for link prediction. It inherits the advances of nonparametric Bayesian methods to infer the unknown latent social dimension, while for discriminative link prediction, it adopts the max-margin learning principle by minimizing a hinge-loss using the linear expectation operator, without dealing with a highly nonlinear link likelihood function. For posterior inference, we develop an efficient stochastic variational inference algorithm under a truncated mean-field assumption. Our methods can scale up to large-scale real networks with millions of entities and tens of millions of positive links. We also provide a full Bayesian formulation, which can avoid tuning regularization hyper-parameters. Experimental results on a diverse range of real datasets demonstrate the benefits inherited from max-margin learning and Bayesian nonparametric inference.

Classifier chains are popular and effective method to tackle a multi-label classification problem. The aim of this paper is to study the asymptotic properties of the chain model in which the conditional probabilities are of the logistic form. In particular we find conditions on the number of labels and the distribution of feature vector under which the estimated mode of the joint distribution of labels converges to the true mode. Best of our knowledge, this important issue has not yet been studied in the context of multi-label learning. We also investigate how the order of model building in a chain influences the estimation of the joint distribution of labels. We establish the link between the problem of incorrect ordering in the chain and incorrect model specification. We propose a procedure of determining the optimal ordering of labels in the chain, which is based on using measures of correct specification and allows to find the ordering such that the consecutive logistic models are best possibly specified. The other important question raised in this paper is how accurately can we estimate the joint posterior probability when the ordering of labels is wrong or the logistic models in the chain are incorrectly specified. The numerical experiments illustrate the theoretical results.