Determining interactions between entities and the overall organization and clustering of nodes in networks is a major challenge when analyzing biological and social network data. Here we extend the Indian Buffet Process (IBP), a nonparametric Bayesian model, to integrate noisy interaction scores with properties of individual entities for inferring interaction networks and clustering nodes within these networks. We present an application of this method to study how microRNAs regulate mRNAs in cells. Analysis of synthetic and real data indicates that the method improves upon prior methods, correctly recovers interactions and clusters, and provides accurate biological predictions.

Inferring key unobservable features of individuals is an important task in the applied sciences. In particular, an important source of data in fields such as marketing, social sciences and medicine is questionnaires: answers in such questionnaires are noisy measures of target unobserved features. While comprehensive surveys help to better estimate the latent variables of interest, aiming at a high number of questions comes at a price: refusal to participate in surveys can go up, as well as the rate of missing data; quality of answers can decline; costs associated with applying such questionnaires can also increase. In this paper, we cast the problem of refining existing models for questionnaire data as follows: solve a constrained optimization problem of preserving the maximum amount of information found in a latent variable model using only a subset of existing questions. The goal is to find an optimal subset of a given size. For that, we first define an information theoretical measure for quantifying the quality of a reduced questionnaire. Three different approximate inference methods are introduced to solve this problem. Comparisons against a simple but powerful heuristic are presented.

Neurons typically respond to a restricted number of stimulus features within the high-dimensional space of natural stimuli. Here we describe an explicit model-based interpretation of traditional estimators for a neuron's multi-dimensional feature space, which allows for several important generalizations and extensions. First, we show that traditional estimators based on the spike-triggered average (STA) and spike-triggered covariance (STC) can be formalized in terms of the "expected log-likelihood" of a Linear-Nonlinear-Poisson (LNP) model with Gaussian stimuli. This model-based formulation allows us to define maximum-likelihood and Bayesian estimators that are statistically consistent and efficient in a wider variety of settings, such as with naturalistic (non-Gaussian) stimuli. It also allows us to employ Bayesian methods for regularization, smoothing, sparsification, and model comparison, and provides Bayesian confidence intervals on model parameters. We describe an empirical Bayes method for selecting the number of features, and extend the model to accommodate an arbitrary elliptical nonlinear response function, which results in a more powerful and more flexible model for feature space inference. We validate these methods using neural data recorded extracellularly from macaque primary visual cortex.

We introduce a variational Bayesian inference algorithm which can be widely applied to sparse linear models. The algorithm is based on the spike and slab prior which, from a Bayesian perspective, is the golden standard for sparse inference. We apply the method to a general multi-task and multiple kernel learning model in which a common set of Gaussian process functions is linearly combined with task-specific sparse weights, thus inducing relation between tasks. This model unifies several sparse linear models, such as generalized linear models, sparse factor analysis and matrix factorization with missing values, so that the variational algorithm can be applied to all these cases. We demonstrate our approach in multi-output Gaussian process regression, multi-class classification, image processing applications and collaborative filtering.

This work considers the problem of learning the structure of multivariate linear tree models, which include a variety of directed tree graphical models with continuous, discrete, and mixed latent variables such as linear-Gaussian models, hidden Markov models, Gaussian mixture models, and Markov evolutionary trees. The setting is one where we only have samples from certain observed variables in the tree, and our goal is to estimate the tree structure (i.e., the graph of how the underlying hidden variables are connected to each other and to the observed variables). We propose the Spectral Recursive Grouping algorithm, an efficient and simple bottom-up procedure for recovering the tree structure from independent samples of the observed variables. Our finite sample size bounds for exact recovery of the tree structure reveal certain natural dependencies on underlying statistical and structural properties of the underlying joint distribution. Furthermore, our sample complexity guarantees have no explicit dependence on the dimensionality of the observed variables, making the algorithm applicable to many high-dimensional settings. At the heart of our algorithm is a spectral quartet test for determining the relative topology of a quartet of variables from second-order statistics.

The L_1 regularized Gaussian maximum likelihood estimator has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to other state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modifications that leverage the structure of the sparse Gaussian MLE problem. We show that our method is superlinearly convergent, and also present experimental results using synthetic and real application data that demonstrate the considerable improvements in performance of our method when compared to other state-of-the-art methods.

A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes' rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric state-space model. A consistency rate for the posterior estimate is established.

Latent variable models are frequently used to identify structure in dichotomous network data, in part because they give rise to a Bernoulli product likelihood that is both well understood and consistent with the notion of exchangeable random graphs. In this article we propose conservative confidence sets that hold with respect to these underlying Bernoulli parameters as a function of any given partition of network nodes, enabling us to assess estimates of \emph{residual} network structure, that is, structure that cannot be explained by known covariates and thus cannot be easily verified by manual inspection. We demonstrate the proposed methodology by analyzing student friendship networks from the National Longitudinal Survey of Adolescent Health that include race, gender, and school year as covariates. We employ a stochastic expectation-maximization algorithm to fit a logistic regression model that includes these explanatory variables as well as a latent stochastic blockmodel component and additional node-specific effects. Although maximum-likelihood estimates do not appear consistent in this context, we are able to evaluate confidence sets as a function of different blockmodel partitions, which enables us to qualitatively assess the significance of estimated residual network structure relative to a baseline, which models covariates but lacks block structure.

There are many settings in which we wish to fit a model of the behavior of individuals but where our data consist only of aggregate information (counts or low-dimensional contingency tables). This paper introduces Collective Graphical Models --a framework for modeling and probabilistic inference that operates directly on the sufficient statistics of the individual model. We derive a highly-efficient Gibbs sampling algorithm for sampling from the posterior distribution of the sufficient statistics conditioned on noisy aggregate observations, prove its correctness, and demonstrate its effectiveness experimentally.

Biased labelers are a systemic problem in crowdsourcing, and a comprehensive toolbox for handling their responses is still being developed. A typical crowdsourcing application can be divided into three steps: data collection, data curation, and learning. At present these steps are often treated separately. We present Bayesian Bias Mitigation for Crowdsourcing (BBMC), a Bayesian model to unify all three. Most data curation methods account for the {\it effects} of labeler bias by modeling all labels as coming from a single latent truth. Our model captures the {\it sources} of bias by describing labelers as influenced by shared random effects. This approach can account for more complex bias patterns that arise in ambiguous or hard labeling tasks and allows us to merge data curation and learning into a single computation. Active learning integrates data collection with learning, but is commonly considered infeasible with Gibbs sampling inference. We propose a general approximation strategy for Markov chains to efficiently quantify the effect of a perturbation on the stationary distribution and specialize this approach to active learning. Experiments show BBMC to outperform many common heuristics.