Solving the Empirical Bayes Normal Means Problem with Correlated Noise

arXiv.org Machine Learning

The Normal Means problem plays a fundamental role in many areas of modern high-dimensional statistics, both in theory and practice. And the Empirical Bayes (EB) approach to solving this problem has been shown to be highly effective, again both in theory and practice. However, almost all EB treatments of the Normal Means problem assume that the observations are independent. In practice correlations are ubiquitous in real-world applications, and these correlations can grossly distort EB estimates. Here, exploiting theory from Schwartzman (2010), we develop new EB methods for solving the Normal Means problem that take account of unknown correlations among observations. We provide practical software implementations of these methods, and illustrate them in the context of large-scale multiple testing problems and False Discovery Rate (FDR) control. In realistic numerical experiments our methods compare favorably with other commonly-used multiple testing methods.


Bayesian Inference of Spreading Processes on Networks

arXiv.org Machine Learning

Infectious diseases are studied to understand their spreading mechanisms, to evaluate control strategies and to predict the risk and course of future outbreaks. Because people only interact with a small number of individuals, and because the structure of these interactions matters for spreading processes, the pairwise relationships between individuals in a population can be usefully represented by a network. Although the underlying processes of transmission are different, the network approach can be used to study the spread of pathogens in a contact network or the spread of rumors in an online social network. We study simulated simple and complex epidemics on synthetic networks and on two empirical networks, a social / contact network in an Indian village and an online social network in the U.S. Our goal is to learn simultaneously about the spreading process parameters and the source node (first infected node) of the epidemic, given a fixed and known network structure, and observations about state of nodes at several points in time. Our inference scheme is based on approximate Bayesian computation (ABC), an inference technique for complex models with likelihood functions that are either expensive to evaluate or analytically intractable. ABC enables us to adopt a Bayesian approach to the problem despite the posterior distribution being very complex. Our method is agnostic about the topology of the network and the nature of the spreading process. It generally performs well and, somewhat counter-intuitively, the inference problem appears to be easier on more heterogeneous network topologies, which enhances its future applicability to real-world settings where few networks have homogeneous topologies.


A Bayesian Group Sparse Multi-Task Regression Model for Imaging Genetics

arXiv.org Machine Learning

Motivation: Recent advances in technology for brain imaging and high-throughput genotyping have motivated studies examining the influence of genetic variation on brain structure. Wang et al. (Bioinformatics, 2012) have developed an approach for the analysis of imaging genomic studies using penalized multi-task regression with regularization based on a novel group $l_{2,1}$-norm penalty which encourages structured sparsity at both the gene level and SNP level. While incorporating a number of useful features, the proposed method only furnishes a point estimate of the regression coefficients; techniques for conducting statistical inference are not provided. A new Bayesian method is proposed here to overcome this limitation. Results: We develop a Bayesian hierarchical modeling formulation where the posterior mode corresponds to the estimator proposed by Wang et al. (Bioinformatics, 2012), and an approach that allows for full posterior inference including the construction of interval estimates for the regression parameters. We show that the proposed hierarchical model can be expressed as a three-level Gaussian scale mixture and this representation facilitates the use of a Gibbs sampling algorithm for posterior simulation. Simulation studies demonstrate that the interval estimates obtained using our approach achieve adequate coverage probabilities that outperform those obtained from the nonparametric bootstrap. Our proposed methodology is applied to the analysis of neuroimaging and genetic data collected as part of the Alzheimer's Disease Neuroimaging Initiative (ADNI), and this analysis of the ADNI cohort demonstrates clearly the value added of incorporating interval estimation beyond only point estimation when relating SNPs to brain imaging endophenotypes.


A rational decision making framework for inhibitory control

Neural Information Processing Systems

Intelligent agents are often faced with the need to choose actions with uncertain consequences, and to modify those actions according to ongoing sensory processing and changing task demands. The requisite ability to dynamically modify or cancel planned actions is known as inhibitory control in psychology. We formalize inhibitory control as a rational decision-making problem, and apply to it to the classical stop-signal task. Using Bayesian inference and stochastic control tools, we show that the optimal policy systematically depends on various parameters of the problem, such as the relative costs of different action choices, the noise level of sensory inputs, and the dynamics of changing environmental demands. Our normative model accounts for a range of behavioral data in humans and animals in the stop-signal task, suggesting that the brain implements statistically optimal, dynamically adaptive, and reward-sensitive decision-making in the context of inhibitory control problems.


Most Relevant Explanation in Bayesian Networks

Journal of Artificial Intelligence Research

A major inference task in Bayesian networks is explaining why some variables are observed in their particular states using a set of target variables. Existing methods for solving this problem often generate explanations that are either too simple (underspecified) or too complex (overspecified). In this paper, we introduce a method called Most Relevant Explanation (MRE) which finds a partial instantiation of the target variables that maximizes the generalized Bayes factor (GBF) as the best explanation for the given evidence. Our study shows that GBF has several theoretical properties that enable MRE to automatically identify the most relevant target variables in forming its explanation. In particular, conditional Bayes factor (CBF), defined as the GBF of a new explanation conditioned on an existing explanation, provides a soft measure on the degree of relevance of the variables in the new explanation in explaining the evidence given the existing explanation. As a result, MRE is able to automatically prune less relevant variables from its explanation. We also show that CBF is able to capture well the explaining-away phenomenon that is often represented in Bayesian networks. Moreover, we define two dominance relations between the candidate solutions and use the relations to generalize MRE to find a set of top explanations that is both diverse and representative. Case studies on several benchmark diagnostic Bayesian networks show that MRE is often able to find explanatory hypotheses that are not only precise but also concise.