We show how smoother priors can preserve the benefits of these sparse priors while adding stability to the Maximum A-Posteriori (MAP) estimate that makes it more useful for prediction problems. Additionally, we show how to calculate the derivative of the MAP estimate efficiently with implicit differentiation. One prior that can be differentiated this way is KL-regularization. We demonstrate its effectiveness on a wide variety of applications, and find that online optimization of the parameters of the KL-regularized model can significantly improve prediction performance.

We explore a new Bayesian model for probabilistic grammars, a family of distributions over discrete structures that includes hidden Markov models and probabilistic context-free grammars. Our model extends the correlated topic model framework to probabilistic grammars, exploiting the logistic normal distribution as a prior over the grammar parameters. We derive a variational EM algorithm for that model, and then experiment with the task of unsupervised grammar induction for natural language dependency parsing. We show that our model achieves superior results over previous models that use different priors.

Stochastic relational models (SRMs) [15] provide a rich family of choices for learning and predicting dyadic data between two sets of entities. The models generalize matrixfactorization to a supervised learning problem that utilizes attributes of entities in a hierarchical Bayesian framework. Previously variational Bayes inference wasapplied for SRMs, which is, however, not scalable when the size of either entity set grows to tens of thousands. In this paper, we introduce a Markov chain Monte Carlo (MCMC) algorithm for equivalent models of SRMs in order to scale the computation to very large dyadic data sets. Both superior scalability and predictive accuracy are demonstrated on a collaborative filtering problem, which involves tens of thousands users and half million items.

Nonparametric Bayesian models provide a framework for flexible probabilistic modelling of complex datasets. Unfortunately, Bayesian inference methods often require high-dimensional averages and can be slow to compute, especially with the potentially unbounded representations associated with nonparametric models. We address the challenge of scaling nonparametric Bayesian inference to the increasingly large datasets found in real-world applications, focusing on the case of parallelising inference in the Indian Buffet Process (IBP). Our approach divides a large data set between multiple processors. The processors use message passing to compute likelihoods in an asynchronous, distributed fashion and to propagate statistics about the global Bayesian posterior. This novel MCMC sampler is the first parallel inference scheme for IBP-based models, scaling to datasets orders of magnitude larger than had previously been possible.

We consider the problem of learning the structure of Ising models (pairwise binary Markov random fields) from i.i.d. samples. While several methods have been proposed to accomplish this task, their relative merits and limitations remain somewhat obscure. By analyzing a number of concrete examples, we show that low-complexity algorithms systematically fail when the Markov random field develops long-range correlations. More precisely, this phenomenon appears to be related to the Ising model phase transition (although it does not coincide with it).

We apply robust Bayesian decision theory to improve both generative and discriminative learners under bias in class proportions in labeled training data, when the true class proportions are unknown. For the generative case, we derive an entropy-based weighting that maximizes expected log likelihood under the worst-case true class proportions. For the discriminative case, we derive a multinomial logistic model that minimizes worst-case conditional log loss. We apply our theory to the modeling of species geographic distributions from presence data, an extreme case of label bias since there is no absence data. On a benchmark dataset, we find that entropy-based weighting offers an improvement over constant estimates of class proportions, consistently reducing log loss on unbiased test data.

The rest of the paper is organized as follows.

We consider the general problem of constructing nonparametric Bayesian models on infinite-dimensional random objects, such as functions, infinite graphs or infinite permutations. The problem has generated much interest in machine learning, where it is treated heuristically, but has not been studied in full generality in nonparametric Bayesian statistics, which tends to focus on models over probability distributions. Our approach applies a standard tool of stochastic process theory, the construction of stochastic processes from their finite-dimensional marginal distributions. The main contribution of the paper is a generalization of the classic Kolmogorov extension theorem to conditional probabilities. This extension allows a rigorous construction of nonparametric Bayesian models from systems of finite-dimensional, parametric Bayes equations. Using this approach, we show (i) how existence of a conjugate posterior for the nonparametric model can be guaranteed by choosing conjugate finite-dimensional models in the construction, (ii) how the mapping to the posterior parameters of the nonparametric model can be explicitly determined, and (iii) that the construction of conjugate models in essence requires the finite-dimensional models to be in the exponential family. As an application of our constructive framework, we derive a model on infinite permutations, the nonparametric Bayesian analogue of a model recently proposed for the analysis of rank data.

The coding of information by neural populations depends critically on the statistical dependencies between neuronal responses. However, there is no simple model that combines the observations that (1) marginal distributions over single-neuron spike counts are often approximately Poisson; and (2) joint distributions over the responses of multiple neurons are often strongly dependent. Here, we show that both marginal and joint properties of neural responses can be captured using Poisson copula models. Copulas are joint distributions that allow random variables with arbitrary marginals to be combined while incorporating arbitrary dependencies between them. Different copulas capture different kinds of dependencies, allowing for a richer and more detailed description of dependencies than traditional summary statistics, such as correlation coefficients. We explore a variety of Poisson copula models for joint neural response distributions, and derive an efficient maximum likelihood procedure for estimating them. We apply these models to neuronal data collected in and macaque motor cortex, and quantify the improvement in coding accuracy afforded by incorporating the dependency structure between pairs of neurons.

The recent emergence of Graphics Processing Units (GPUs) as general-purpose parallel computing devices provides us with new opportunities to develop scalable learning methods for massive data. In this work, we consider the problem of parallelizing two inference methods on GPUs for latent Dirichlet Allocation (LDA) models, collapsed Gibbs sampling (CGS) and collapsed variational Bayesian (CVB). To address limited memory constraints on GPUs, we propose a novel data partitioning scheme that effectively reduces the memory cost. Furthermore, the partitioning scheme balances the computational cost on each multiprocessor and enables us to easily avoid memory access conflicts. We also use data streaming to handle extremely large datasets. Extensive experiments showed that our parallel inference methods consistently produced LDA models with the same predictive power as sequential training methods did but with 26x speedup for CGS and 196x speedup for CVB on a GPU with 30 multiprocessors; actually the speedup is almost linearly scalable with the number of multiprocessors available. The proposed partitioning scheme and data streaming can be easily ported to many other models in machine learning.