Modern genomic studies are increasingly focused on discovering more and more interesting genes associated with a health response. Traditional shrinkage priors are primarily designed to detect a handful of signals from tens and thousands of predictors
. Under diverse sparsity regimes, the nature of signal detection is associated with a tail behaviour of a prior. A desirable tail behaviour is called tail-adaptive shrinkage property where tail-heaviness of a prior gets adaptively larger (or smaller) as a sparsity level increases (or decreases) to accommodate more (or less) signals. We propose a global-local-tail (GLT) Gaussian mixture distribution to ensure this property and provide accurate inference under diverse sparsity regimes. Incorporating a peaks-over-threshold method in extreme value theory, we develop an automated tail learning algorithm for the GLT prior. We compare the performance of the GLT prior to the Horseshoe in two gene expression datasets and numerical examples. Results suggest that varying tail rule is advantageous over fixed tail rule under diverse sparsity domains.
Estimating the marginal and joint densities of the long-term average intakes of different dietary components is an important problem in nutritional epidemiology. Since these variables cannot be directly measured, data are usually collected in the for
m of 24-hour recalls of the intakes, which show marked patterns of conditional heteroscedasticity. Significantly compounding the challenges, the recalls for episodically consumed dietary components also include exact zeros. The problem of estimating the density of the latent long-time intakes from their observed measurement error contaminated proxies is then a problem of deconvolution of densities with zero-inflated data. We propose a Bayesian semiparametric solution to the problem, building on a novel hierarchical latent variable framework that translates the problem to one involving continuous surrogates only. Crucial to accommodating important aspects of the problem, we then design a copula-based approach to model the involved joint distributions, adopting different modeling strategies for the marginals of the different dietary components. We design efficient Markov chain Monte Carlo algorithms for posterior inference and illustrate the efficacy of the proposed method through simulation experiments. Applied to our motivating nutritional epidemiology problems, compared to other approaches, our method provides more realistic estimates of the consumption patterns of episodically consumed dietary components.
We consider the problem of multivariate density deconvolution when the interest lies in estimating the distribution of a vector-valued random variable but precise measurements of the variable of interest are not available, observations being contamin
ated with additive measurement errors. The existing sparse literature on the problem assumes the density of the measurement errors to be completely known. We propose robust Bayesian semiparametric multivariate deconvolution approaches when the measurement error density is not known but replicated proxies are available for each unobserved value of the random vector. Additionally, we allow the variability of the measurement errors to depend on the associated unobserved value of the vector of interest through unknown relationships which also automatically includes the case of multivariate multiplicative measurement errors. Basic properties of finite mixture models, multivariate normal kernels and exchangeable priors are exploited in many novel ways to meet the modeling and computational challenges. Theoretical results that show the flexibility of the proposed methods are provided. We illustrate the efficiency of the proposed methods in recovering the true density of interest through simulation experiments. The methodology is applied to estimate the joint consumption pattern of different dietary components from contaminated 24 hour recalls.
