An effective strategy to exploit the supervising side information for discovering predictive topic representations is to impose discriminative constraints induced by such information on the posterior distributions under a topic model. This strategy has been adopted by a number of supervised topic models, such as MedLDA, which employs max-margin posterior constraints. However, unlike the likelihood-based supervised topic models, of which posterior inference can be carried out using the Bayes' rule, the max-margin posterior constraints have made Monte Carlo methods infeasible or at least not directly applicable, thereby limited the choice of inference algorithms to be based on variational approximation with strict mean field assumptions. In this paper, we develop two efficient Monte Carlo methods under much weaker assumptions for max-margin supervised topic models based on an importance sampler and a collapsed Gibbs sampler, respectively, in a convex dual formulation. We report thorough experimental results that compare our approach favorably against existing alternatives in both accuracy and efficiency.

Sparse graphical modelling/inverse covariance selection is an important problem in machine learning and has seen significant advances in recent years. A major focus has been on methods which perform model selection in high dimensions. To this end, numerous convex $\ell_1$ regularization approaches have been proposed in the literature. It is not however clear which of these methods are optimal in any well-defined sense. A major gap in this regard pertains to the rate of convergence of proposed optimization methods. To address this, an iterative thresholding algorithm for numerically solving the $\ell_1$-penalized maximum likelihood problem for sparse inverse covariance estimation is presented. The proximal gradient method considered in this paper is shown to converge at a linear rate, a result which is the first of its kind for numerically solving the sparse inverse covariance estimation problem. The convergence rate is provided in closed form, and is related to the condition number of the optimal point. Numerical results demonstrating the proven rate of convergence are presented.

A common challenge for Bayesian models of perception is the fact that the two fundamental Bayesian components, the prior distribution and the likelihood function, areformally unconstrained. Here we argue that a neural system that emulates Bayesian inference is naturally constrained by the way it represents sensory information inpopulations of neurons. More specifically, we show that an efficient coding principle creates a direct link between prior and likelihood based on the underlying stimulus distribution. The resulting Bayesian estimates can show biases awayfrom the peaks of the prior distribution, a behavior seemingly at odds with the traditional view of Bayesian estimation, yet one that has been reported in human perception. We demonstrate that our framework correctly accounts for the repulsive biases previously reported for the perception of visual orientation, and show that the predicted tuning characteristics of the model neurons match the reported orientation tuning properties of neurons in primary visual cortex. Our results suggest that efficient coding is a promising hypothesis in constraining Bayesianmodels of perceptual inference.

We present a novel marginalized particle Gaussian process (MPGP) regression, which provides a fast, accurate online Bayesian filtering framework to model the latent function. Using a state space model established by the data construction procedure, our MPGP recursively filters out the estimation of hidden function values by a Gaussian mixture. Meanwhile, it provides a new online method for training hyperparameters with a number of weighted particles. We demonstrate the estimated performance of our MPGP on both simulated and real large data sets. The results show that our MPGP is a robust estimation algorithm with high computational efficiency, which outperforms other state-of-art sparse GP methods.

We consider the problem of learning control policies via trajectory preference queries to an expert. In particular, the learning agent can present an expert with short runs of a pair of policies originating from the same state and the expert then indicates the preferred trajectory. The agent's goal is to elicit a latent target policy from the expert with as few queries as possible. To tackle this problem we propose a novel Bayesian model of the querying process and introduce two methods that exploit this model to actively select expert queries. Experimental results on four benchmark problems indicate that our model can effectively learn policies from trajectory preference queries and that active query selection can be substantially more efficient than random selection.

We present a new learning strategy based on an efficient blocked Gibbs sampler for sparse overcomplete linear models. Particular emphasis is placed on statistical image modeling, where overcomplete models have played an important role in discovering sparse representations. Our Gibbs sampler is faster than general purpose sampling schemes while also requiring no tuning as it is free of parameters. Using the Gibbs sampler and a persistent variant of expectation maximization, we are able to extract highly sparse distributions over latent sources from data. When applied to natural images, our algorithm learns source distributions which resemble spike-and-slab distributions. We evaluate the likelihood and quantitatively compare the performance of the overcomplete linear model to its complete counterpart as well as a product of experts model, which represents another overcomplete generalization of the complete linear model. In contrast to previous claims, we find that overcomplete representations lead to significant improvements, but that the overcomplete linear model still underperforms other models.

The variational Bayesian (VB) approach is one of the best tractable approximations to the Bayesian estimation, and it was demonstrated to perform well in many applications. However, its good performance was not fully understood theoretically. For example, VB sometimes produces a sparse solution, which is regarded as a practical advantage of VB, but such sparsity is hardly observed in the rigorous Bayesian estimation. In this paper, we focus on probabilistic PCA and give more theoretical insight into the empirical success of VB. More specifically, for the situation where the noise variance is unknown, we derive a sufficient condition for perfect recovery of the true PCA dimensionality in the large-scale limit when the size of an observed matrix goes to infinity. In our analysis, we obtain bounds for a noise variance estimator and simple closed-form solutions for other parameters, which themselves are actually very useful for better implementation of VB-PCA.

The empirical success of the belief propagation approximate inference algorithm has inspired numerous theoretical and algorithmic advances. Yet, for continuous non-Gaussian domains performing belief propagation remains a challenging task: recent innovations such as nonparametric or kernel belief propagation, while useful, come with a substantial computational cost and offer little theoretical guarantees, even for tree structured models. In this work we present Nonparanormal BP for performing efficient inference on distributions parameterized by a Gaussian copulas network and any univariate marginals. For tree structured networks, our approach is guaranteed to be exact for this powerful class of non-Gaussian models. Importantly, the method is as efficient as standard Gaussian BP, and its convergence properties do not depend on the complexity of the univariate marginals, even when a nonparametric representation is used.

The partition function plays a key role in probabilistic modeling including conditional random fields, graphical models, and maximum likelihood estimation. To optimize partition functions, this article introduces a quadratic variational upper bound. This inequality facilitates majorization methods: optimization of complicated functions through the iterative solution of simpler sub-problems. Such bounds remain efficient to compute even when the partition function involves a graphical model (with small tree-width) or in latent likelihood settings. For large-scale problems, low-rank versions of the bound are provided and outperform LBFGS as well as first-order methods. Several learning applications are shown and reduce to fast and convergent update rules. Experimental results show advantages over state-of-the-art optimization methods.

Learning temporal dependencies between variables over continuous time is an important and challenging task. Continuous-time Bayesian networks effectively model such processes but are limited by the number of conditional intensity matrices, which grows exponentially in the number of parents per variable. We develop a partition-based representation using regression trees and forests whose parameter spaces grow linearly in the number of node splits. Using a multiplicative assumption we show how to update the forest likelihood in closed form, producing efficient model updates. Our results show multiplicative forests can be learned from few temporal trajectories with large gains in performance and scalability.