In this article, we investigate large sample properties of model selection procedures in a general Bayesian framework when a closed form expression of the marginal likelihood function is not available or a local asymptotic quadratic approximation of the log-likelihood function does not exist. Under appropriate identifiability assumptions on the true model, we provide sufficient conditions for a Bayesian model selection procedure to be consistent and obey the Occam's razor phenomenon, i.e., the probability of selecting the "smallest" model that contains the truth tends to one as the sample size goes to infinity. In order to show that a Bayesian model selection procedure selects the smallest model containing the truth, we impose a prior anti-concentration condition, requiring the prior mass assigned by large models to a neighborhood of the truth to be sufficiently small. In a more general setting where the strong model identifiability assumption may not hold, we introduce the notion of local Bayesian complexity and develop oracle inequalities for Bayesian model selection procedures. Our Bayesian oracle inequality characterizes a trade-off between the approximation error and a Bayesian characterization of the local complexity of the model, illustrating the adaptive nature of averaging-based Bayesian procedures towards achieving an optimal rate of posterior convergence. Specific applications of the model selection theory are discussed in the context of high-dimensional nonparametric regression and density regression where the regression function or the conditional density is assumed to depend on a fixed subset of predictors. As a result of independent interest, we propose a general technique for obtaining upper bounds of certain small ball probability of stationary Gaussian processes.

In many real applications of statistical learning, a decision made from misclassification can be too costly to afford; in this case, a reject option, which defers the decision until further investigation is conducted, is often preferred. In recent years, there has been much development for binary classification with a reject option. Yet, little progress has been made for the multicategory case. In this article, we propose margin-based multicategory classification methods with a reject option. In addition, and more importantly, we introduce a new and unique refine option for the multicategory problem, where the class of an observation is predicted to be from a set of class labels, whose cardinality is not necessarily one. The main advantage of both options lies in their capacity of identifying error-prone observations. Moreover, the refine option can provide more constructive information for classification by effectively ruling out implausible classes. Efficient implementations have been developed for the proposed methods. On the theoretical side, we offer a novel statistical learning theory and show a fast convergence rate of the excess $\ell$-risk of our methods with emphasis on diverging dimensionality and number of classes. The results can be further improved under a low noise assumption. A set of comprehensive simulation and real data studies has shown the usefulness of the new learning tools compared to regular multicategory classifiers. Detailed proofs of theorems and extended numerical results are included in the supplemental materials available online.

We present a general framework, the coupled compound Poisson factorization (CCPF), to capture the missing-data mechanism in extremely sparse data sets by coupling a hierarchical Poisson factorization with an arbitrary data-generating model. We derive a stochastic variational inference algorithm for the resulting model and, as examples of our framework, implement three different data-generating models---a mixture model, linear regression, and factor analysis---to robustly model non-random missing data in the context of clustering, prediction, and matrix factorization. In all three cases, we test our framework against models that ignore the missing-data mechanism on large scale studies with non-random missing data, and we show that explicitly modeling the missing-data mechanism substantially improves the quality of the results, as measured using data log likelihood on a held-out test set.

Processes such as disease propagation and information diffusion often spread over some latent network structure which must be learned from observation. Given a set of unlabeled training examples representing occurrences of an event type of interest (e.g., a disease outbreak), our goal is to learn a graph structure that can be used to accurately detect future events of that type. Motivated by new theoretical results on the consistency of constrained and unconstrained subset scans, we propose a novel framework for learning graph structure from unlabeled data by comparing the most anomalous subsets detected with and without the graph constraints. Our framework uses the mean normalized log-likelihood ratio score to measure the quality of a graph structure, and efficiently searches for the highest-scoring graph structure. Using simulated disease outbreaks injected into real-world Emergency Department data from Allegheny County, we show that our method learns a structure similar to the true underlying graph, but enables faster and more accurate detection.

We proposed a probabilistic approach to joint modeling of participants' reliability and humans' regularity in crowdsourced affective studies. Reliability measures how likely a subject will respond to a question seriously; and regularity measures how often a human will agree with other seriously-entered responses coming from a targeted population. Crowdsourcing-based studies or experiments, which rely on human self-reported affect, pose additional challenges as compared with typical crowdsourcing studies that attempt to acquire concrete non-affective labels of objects. The reliability of participants has been massively pursued for typical non-affective crowdsourcing studies, whereas the regularity of humans in an affective experiment in its own right has not been thoroughly considered. It has been often observed that different individuals exhibit different feelings on the same test question, which does not have a sole correct response in the first place. High reliability of responses from one individual thus cannot conclusively result in high consensus across individuals. Instead, globally testing consensus of a population is of interest to investigators. Built upon the agreement multigraph among tasks and workers, our probabilistic model differentiates subject regularity from population reliability. We demonstrate the method's effectiveness for in-depth robust analysis of large-scale crowdsourced affective data, including emotion and aesthetic assessments collected by presenting visual stimuli to human subjects.

Computation of moments of transformed random variables is a problem appearing in many engineering applications. The current methods for moment transformation are mostly based on the classical quadrature rules which cannot account for the approximation errors. Our aim is to design a method for moment transformation for Gaussian random variables which accounts for the error in the numerically computed mean. We employ an instance of Bayesian quadrature, called Gaussian process quadrature (GPQ), which allows us to treat the integral itself as a random variable, where the integral variance informs about the incurred integration error. Experiments on the coordinate transformation and nonlinear filtering examples show that the proposed GPQ moment transform performs better than the classical transforms.

Complex activity recognition is challenging due to the inherent uncertainty and diversity of performing a complex activity. Normally, each instance of a complex activity has its own configuration of atomic actions and their temporal dependencies. We propose in this paper an atomic action-based Bayesian model that constructs Allen's interval relation networks to characterize complex activities with structural varieties in a probabilistic generative way: By introducing latent variables from the Chinese restaurant process, our approach is able to capture all possible styles of a particular complex activity as a unique set of distributions over atomic actions and relations. We also show that local temporal dependencies can be retained and are globally consistent in the resulting interval network. Moreover, network structure can be learned from empirical data. A new dataset of complex hand activities has been constructed and made publicly available, which is much larger in size than any existing datasets. Empirical evaluations on benchmark datasets as well as our in-house dataset demonstrate the competitiveness of our approach.

Testing whether a probability distribution is compatible with a given Bayesian network is a fundamental task in the field of causal inference, where Bayesian networks model causal relations. Here we consider the class of causal structures where all correlations between observed quantities are solely due to the influence from latent variables. We show that each model of this type imposes a certain signature on the observable covariance matrix in terms of a particular decomposition into positive semidefinite components. This signature, and thus the underlying hypothetical latent structure, can be tested in a computationally efficient manner via semidefinite programming. This stands in stark contrast with the algebraic geometric tools required if the full observable probability distribution is taken into account. The semidefinite test is compared with tests based on entropic inequalities.

Sparse Bayesian learning is one of the state-of- the-art machine learning algorithms, which is able to make stable and reliable probabilistic predictions. However, some of these algorithms, e.g. probabilistic classification vector machine (PCVM) and relevant vector machine (RVM), are not capable of eliminating irrelevant and redundant features which could lead to performance degradation. To tackle this problem, in this paper, we propose a sparse Bayesian classifier which simultaneously selects the relevant samples and features. We name this classifier a probabilistic feature selection and classification vector machine (PFCVM), in which truncated Gaussian distributions are em- ployed as both sample and feature priors. In order to derive the analytical solution for the proposed algorithm, we use Laplace approximation to calculate approximate posteriors and marginal likelihoods. Finally, we obtain the optimized parameters and hyperparameters by the type-II maximum likelihood method. The experiments on synthetic data set, benchmark data sets and high dimensional data sets validate the performance of PFCVM under two criteria: accuracy of classification and efficacy of selected features. Finally, we analyze the generalization performance of PFCVM and derive a generalization error bound for PFCVM. Then by tightening the bound, we demonstrate the significance of the sparseness for the model.

We propose a method for recovering the structure of a sparse undirected graphical model when very few samples are available. The method decides about the presence or absence of bonds between pairs of variable by considering one pair at a time and using a closed form formula, analytically derived by calculating the posterior probability for every possible model explaining a two body system using Jeffreys prior. The approach does not rely on the optimisation of any cost functions and consequently is much faster than existing algorithms. Despite this time and computational advantage, numerical results show that for several sparse topologies the algorithm is comparable to the best existing algorithms, and is more accurate in the presence of hidden variables. We apply this approach to the analysis of US stock market data and to neural data, in order to show its efficiency in recovering robust statistical dependencies in real data with non stationary correlations in time and space.