We study the convergence rate of stochastic optimization of exact (NP-hard) objectives, for which only biased estimates of the gradient are available. We motivate this problem in the context of learning the structure and parameters of Ising models. We first provide a convergence-rate analysis of deterministic errors for forward-backward splitting (FBS). We then extend our analysis to biased stochastic errors, by first characterizing a family of samplers and providing a high probability bound that allows understanding not only FBS, but also proximal gradient (PG) methods. We derive some interesting conclusions: FBS requires only a logarithmically increasing number of random samples in order to converge (although at a very low rate); the required number of random samples is the same for the deterministic and the biased stochastic setting for FBS and basic PG; accelerated PG is not guaranteed to converge in the biased stochastic setting.

We present a framework for online inference in the presence of a nonexhaustively defined set of classes that incorporates supervised classification with class discovery and modeling. A Dirichlet process prior (DPP) model defined over class distributions ensures that both known and unknown class distributions originate according to a common base distribution. In an attempt to automatically discover potentially interesting class formations, the prior model is coupled with a suitably chosen data model, and sequential Monte Carlo sampling is used to perform online inference. Our research is driven by a biodetection application, where a new class of pathogen may suddenly appear, and the rapid increase in the number of samples originating from this class indicates the onset of an outbreak.

Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data variance is already partially explained by other factors, for example sparse conditional dependencies between the covariates, or temporal correlations leaving some residual variance. We decompose the residual variance into its components through a generalised eigenvalue problem, which we call residual component analysis (RCA). We explore a range of new algorithms that arise from the framework, including one that factorises the covariance of a Gaussian density into a low-rank and a sparse-inverse component. We illustrate the ideas on the recovery of a protein-signaling network, a gene expression time-series data set and the recovery of the human skeleton from motion capture 3-D cloud data.

This paper revisits the problem of analyzing multiple ratings given by different judges. Different from previous work that focuses on distilling the true labels from noisy crowdsourcing ratings, we emphasize gaining diagnostic insights into our in-house well-trained judges. We generalize the well-known DawidSkene model (Dawid & Skene, 1979) to a spectrum of probabilistic models under the same "TrueLabel + Confusion" paradigm, and show that our proposed hierarchical Bayesian model, called HybridConfusion, consistently outperforms DawidSkene on both synthetic and real-world data sets.

Chordal graphs can be used to encode dependency models that are representable by both directed acyclic and undirected graphs. This paper discusses a very simple and efficient algorithm to learn the chordal structure of a probabilistic model from data. The algorithm is a greedy hill-climbing search algorithm that uses the inclusion boundary neighborhood over chordal graphs. In the limit of a large sample size and under appropriate hypotheses on the scoring criterion, we prove that the algorithm will find a structure that is inclusion-optimal when the dependency model of the data-generating distribution can be represented exactly by an undirected graph. The algorithm is evaluated on simulated datasets.

The paper introduces AND/OR importance sampling for probabilistic graphical models. In contrast to importance sampling, AND/OR importance sampling caches samples in the AND/OR space and then extracts a new sample mean from the stored samples. We prove that AND/OR importance sampling may have lower variance than importance sampling; thereby providing a theoretical justification for preferring it over importance sampling. Our empirical evaluation demonstrates that AND/OR importance sampling is far more accurate than importance sampling in many cases.

It is well-known that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with unbounded treewidth in which inference is tractable? Subject to a combinatorial hypothesis due to Robertson et al. (1994), we show that low treewidth is indeed the only structural restriction that can ensure tractability. Thus, even for the "best case" graph structure, there is no inference algorithm with complexity polynomial in the treewidth.

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.

In the Bayesian approach to structure learning of graphical models, the equivalent sample size (ESS) in the Dirichlet prior over the model parameters was recently shown to have an important effect on the maximum-a-posteriori estimate of the Bayesian network structure. In our first contribution, we theoretically analyze the case of large ESS-values, which complements previous work: among other results, we find that the presence of an edge in a Bayesian network is favoured over its absence even if both the Dirichlet prior and the data imply independence, as long as the conditional empirical distribution is notably different from uniform. In our second contribution, we focus on realistic ESS-values, and provide an analytical approximation to the "optimal" ESS-value in a predictive sense (its accuracy is also validated experimentally): this approximation provides an understanding as to which properties of the data have the main effect determining the "optimal" ESS-value.

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.