Variational Bayesian inference and (collapsed) Gibbs sampling are the two important classes of inference algorithms for Bayesian networks. Both have their advantages and disadvantages: collapsed Gibbs sampling is unbiased but is also inefficient for large count values and requires averaging over many samples to reduce variance. On the other hand, variational Bayesian inference is efficient and accurate for large count values but suffers from bias for small counts. We propose a hybrid algorithm that combines the best of both worlds: it samples very small counts and applies variational updates to large counts. This hybridization is shown to significantly improve testset perplexity relative to variational inference at no computational cost.

Exemplar-based clustering methods have been shown to produce state-of-the-art results on a number of synthetic and real-world clustering problems. They are appealing because they offer computational benefits over latent-mean models and can handle arbitrary pairwise similarity measures between data points. However, when trying to recover underlying structure in clustering problems, tailored similarity measures are often not enough; we also desire control over the distribution of cluster sizes. Priors such as Dirichlet process priors allow the number of clusters to be unspecified while expressing priors over data partitions. To our knowledge, they have not been applied to exemplar-based models. We show how to incorporate priors, including Dirichlet process priors, into the recently introduced affinity propagation algorithm. We develop an efficient maxproduct belief propagation algorithm for our new model and demonstrate experimentally how the expanded range of clustering priors allows us to better recover true clusterings in situations where we have some information about the generating process.

Much recent work has concerned sparse approximations to speed up the Gaussian process regression from the unfavorable O(n3) scaling in computational time to O(nm2). Thus far, work has concentrated on models with one covariance function. However, in many practical situations additive models with multiple covariance functions may perform better, since the data may contain both long and short length-scale phenomena. The long length-scales can be captured with global sparse approximations, such as fully independent conditional (FIC), and the short length-scales can be modeled naturally by covariance functions with compact support (CS). CS covariance functions lead to naturally sparse covariance matrices, which are computationally cheaper to handle than full covariance matrices. In this paper, we propose a new sparse Gaussian process model with two additive components: FIC for the long length-scales and CS covariance function for the short length-scales. We give theoretical and experimental results and show that under certain conditions the proposed model has the same computational complexity as FIC. We also compare the model performance of the proposed model to additive models approximated by fully and partially independent conditional (PIC). We use real data sets and show that our model outperforms FIC and PIC approximations for data sets with two additive phenomena.

Nonparametric Bayesian models are often based on the assumption that the objects being modeled are exchangeable. While appropriate in some applications (e.g., bag-of-words models for documents), exchangeability is sometimes assumed simply for computational reasons; non-exchangeable models might be a better choice for applications based on subject matter. Drawing on ideas from graphical models and phylogenetics, we describe a non-exchangeable prior for a class of nonparametric latent feature models that is nearly as efficient computationally as its exchangeable counterpart. Our model is applicable to the general setting in which the dependencies between objects can be expressed using a tree, where edge lengths indicate the strength of relationships. We demonstrate an application to modeling probabilistic choice.

Confidence measures for the generalization error are crucial when small training samples are used to construct classifiers. A common approach is to estimate the generalization error by resampling and then assume the resampled estimator follows a known distribution to form a confidence set [Kohavi 1995, Martin 1996,Yang 2006]. Alternatively, one might bootstrap the resampled estimator of the generalization error to form a confidence set. Unfortunately, these methods do not reliably provide sets of the desired confidence. The poor performance appears to be due to the lack of smoothness of the generalization error as a function of the learned classifier. This results in a non-normal distribution of the estimated generalization error. We construct a confidence set for the generalization error by use of a smooth upper bound on the deviation between the resampled estimate and generalization error. The confidence set is formed by bootstrapping this upper bound. In cases in which the approximation class for the classifier can be represented as a parametric additive model, we provide a computationally efficient algorithm. This method exhibits superior performance across a series of test and simulated data sets.

Inference problems in graphical models can be represented as a constrained optimization of a free energy function. It is known that when the Bethe free energy is used, the fixedpoints of the belief propagation (BP) algorithm correspond to the local minima of the free energy. However BP fails to converge in many cases of interest. Moreover, the Bethe free energy is non-convex for graphical models with cycles thus introducing great difficulty in deriving efficient algorithms for finding local minima of the free energy for general graphs. In this paper we introduce two efficient BP-like algorithms, one sequential and the other parallel, that are guaranteed to converge to the global minimum, for any graph, over the class of energies known as "convex free energies". In addition, we propose an efficient heuristic for setting the parameters of the convex free energy based on the structure of the graph.

An important task in data analysis is the discovery of causal relationships between observed variables. For continuous-valued data, linear acyclic causal models are commonly used to model the data-generating process, and the inference of such models is a well-studied problem. However, existing methods have significant limitations. Methods based on conditional independencies (Spirtes et al. 1993; Pearl 2000) cannot distinguish between independence-equivalent models, whereas approaches purely based on Independent Component Analysis (Shimizu et al. 2006) are inapplicable to data which is partially Gaussian. In this paper, we generalize and combine the two approaches, to yield a method able to learn the model structure in many cases for which the previous methods provide answers that are either incorrect or are not as informative as possible. We give exact graphical conditions for when two distinct models represent the same family of distributions, and empirically demonstrate the power of our method through thorough simulations.

We introduce a new type of graphical model called a "cumulative distribution network" (CDN), which expresses a joint cumulative distribution as a product of local functions. Each local function can be viewed as providing evidence about possible orderings, or rankings, of variables. Interestingly, we find that the conditional independence properties of CDNs are quite different from other graphical models. We also describe a messagepassing algorithm that efficiently computes conditional cumulative distributions. Due to the unique independence properties of the CDN, these messages do not in general have a one-to-one correspondence with messages exchanged in standard algorithms, such as belief propagation. We demonstrate the application of CDNs for structured ranking learning using a previously-studied multi-player gaming dataset.

Parameter estimation in Markov random fields (MRFs) is a difficult task, in which inference over the network is run in the inner loop of a gradient descent procedure. Replacing exact inference with approximate methods such as loopy belief propagation (LBP) can suffer from poor convergence. In this paper, we provide a different approach for combining MRF learning and Bethe approximation. We consider the dual of maximum likelihood Markov network learning - maximizing entropy with moment matching constraints - and then approximate both the objective and the constraints in the resulting optimization problem. Unlike previous work along these lines (Teh & Welling, 2003), our formulation allows parameter sharing between features in a general log-linear model, parameter regularization and conditional training. We show that piecewise training (Sutton & McCallum, 2005) is a very restricted special case of this formulation. We study two optimization strategies: one based on a single convex approximation and one that uses repeated convex approximations. We show results on several real-world networks that demonstrate that these algorithms can significantly outperform learning with loopy and piecewise. Our results also provide a framework for analyzing the trade-offs of different relaxations of the entropy objective and of the constraints.

Latent topic models have been successfully applied as an unsupervised topic discovery technique in large document collections. With the proliferation of hypertext document collection such as the Internet, there has also been great interest in extending these approaches to hypertext [6, 9]. These approaches typically model links in an analogous fashion to how they model words - the document-link co-occurrence matrix is modeled in the same way that the document-word co-occurrence matrix is modeled in standard topic models. In this paper we present a probabilistic generative model for hypertext document collections that explicitly models the generation of links. Specifically, links from a word w to a document d depend directly on how frequent the topic of w is in d, in addition to the in-degree of d. We show how to perform EM learning on this model efficiently. By not modeling links as analogous to words, we end up using far fewer free parameters and obtain better link prediction results.