In this paper we consider the problem of multiple testing when the hypotheses are dependent. In most of the existing literature, either Bayesian or non-Bayesian, the decision rules mainly focus on the validity of the test procedure rather than actually utilizing the dependency to increase efficiency. Moreover, the decisions regarding different hypotheses are marginal in the sense that they do not depend upon each other directly. However, in realistic situations, the hypotheses are usually dependent, and hence it is desirable that the decisions regarding the dependent hypotheses are taken jointly.In this article we develop a novel Bayesian multiple testing procedure that coherently takes this requirement into consideration. Our method, which is based on new notions of error and non-error terms, substantially enhances efficiency by judicious exploitation of the dependence structure among the hypotheses. We prove that our method minimizes the posterior expected loss associated with a an additive "0-1" loss function; we also prove theoretical results on the relevant error probabilities, establishing the coherence and usefulness of our method. The optimal decision configuration is not available in closed form and we propose a novel and efficient simulated annealing algorithm for the purpose of optimization, which is also generically applicable to binary optimization problems.Numerical studies demonstrate that in dependent situations, our method performs significantly better than some existing popular conventional multiple testing methods, in terms of accuracy and power control. Moreover, application of our ideas to a real, spatial data set associated with radionuclide concentration in Rongelap islands yielded insightful results. arises in many multiple testing scenarios. For example, in spatial data where the geographical locations are nearby, the test statistics for different hypotheses are quite likely to be influenced by each other. In microarray experiments, different genes may cluster into groups along biological pathways and exhibit high correlation. In public health studies, the observed data from different time periods and locations are often serially or spatially correlated. Benjamini and Yekutieli (2001) have shown that control over FDR is achieved for certain kinds of positive dependency among the tests. Finner et al. (2002, 2007); Efron (2007) discussed the effect of dependence among test statistics, among others. Qiu et al. (2005) showed that dependence among test statistics significantly affects the power of many F DR controlling procedures. Schwartzman and Lin (2011) and Fan et al. (2012) discussed estimation of F DR under correlation.However, in both classical and Bayesian literature, even in the dependent set-ups, most of the methods are concerned with marginal decision rules, in the sense that the decisions mainly depend upon the marginal distributions of the test statistics, marginal p-values or marginal posterior probabilities. In cases where we have additional information about dependency among t...

The effect of dependence among multiple hypothesis testing have recently attracted attention of the statistical community. Statisticians have studied the effect of dependence in multiple testing procedures under different setups. In this article, we study asymptotic properties of Bayesian multiple testing procedures. Specifically, we provide sufficient conditions for strong consistency under general dependence structure. We also consider a novel Bayesian non-marginal multiple testing procedure and associated error measures that coherently account for the dependence structure present in the model and the prior.We advocate posterior versions of F DR and F N R as appropriate error rates in Bayesian multiple testing and show that the asymptotic convergence rates of the error rates are directly associated with the Kullback-Leibler divergence from the true model. Indeed, all our results hold very generally even under the setup where the class of postulated models is misspecified.We illustrate our general asymptotic theory in a time-varying covariate selection problem with autoregressive response variables, viewed from a multiple testing perspective. We show that for any proper prior distribution on the parameters, consistency of certain Bayesian multiple testing procedures hold.We compare the Bayesian non-marginal procedure with some existing Bayesian multiple testing methods through an extensive simulation study in the above time-varying covariate selection problem. Superior performance of the new procedure compared to the others vindicate that proper exploitation of the dependence structure by the multiple testing methods is indeed important.

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Gaussian graphical models are popular tools for studying the dependence relationships between different random variables. We propose a novel approach to Gaussian graphical models that relies on decomposing the precision matrix encoding the conditional independence relationships into a low rank and a diagonal component. Such decompositions are already popular for modeling large covariance matrices as they admit a latent factor based representation that allows easy inference but are yet to garner widespread use in precision matrix models due to their computational intractability. We show that a simple latent variable representation for such decomposition in fact exists for precision matrices as well. The latent variable construction provides fundamentally novel insights into Gaussian graphical models. It is also immediately useful in Bayesian settings in achieving efficient posterior inference via a straightforward Gibbs sampler that scales very well to high-dimensional problems far beyond the limits of the current state-of-the-art. The ability to efficiently explore the full posterior space allows the model uncertainty to be easily assessed and the underlying graph to be determined via a novel posterior false discovery rate control procedure. The decomposition also crucially allows us to adapt sparsity inducing priors to shrink insignificant off-diagonal entries toward zero, making the approach adaptable to highdimensional small-sample-size sparse settings. We evaluate the method's empirical performance through synthetic experiments and illustrate its practical utility in data sets from two different application domains.

Iodine intake of breast-fed infants was at the limit of above requirement, and they are possibly at the risk of excess iodine intake. In iodine deficient and excessive iodine intake mothers, their infants' serum FT and TSH are independent on their iodine nutritional status but dependent on thyroid hormone profile of their mothers but differently. What is Known: • A median urinary iodine of 100 μg/L is used to define adequate iodine intake of lactating mothers and children < 2 years. However, adequate iodine intake in terms of urinary iodine of infants of age 1-3 months is not known. What is New: • Iodine intake of absolutely breast-fed infants (1-3 months) was more than adequate, though their mother's intake was adequate as breast milk contains more iodine than urine. The infants of iodine deficient and excessive iodine intake mothers, infants' hormonal profile is independent of their iodine nutritional status but dependent on their mothers thyroid hormone profile.