  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 form of 24-hour recalls of the intakes. The problem of
estimating the density of the latent long-term average intakes from their
observed but error contaminated recalls then becomes a problem of multivariate
deconvolution of densities. The underlying densities could potentially vary
with the subjects' demographic characteristics such as sex, ethnicity, age,
etc. The problem of density deconvolution in the presence of associated
precisely measured covariates has, however, never been considered before, not
even in the univariate setting. We present a flexible Bayesian semiparametric
approach to covariate informed multivariate deconvolution. Building on recent
advances in copula deconvolution and conditional tensor factorization
techniques, our proposed method not only allows the joint and the marginal
densities to vary flexibly with the associated predictors but also allows
automatic selection of the most influential predictors. Importantly, the method
also allows the density of interest and the density of the measurement errors
to vary with potentially different sets of predictors. We design Markov chain
Monte Carlo algorithms that enable efficient posterior inference, appropriately
accommodating uncertainty in all aspects of our analysis. The empirical
efficacy of the proposed method is illustrated through simulation experiments.
Its practical utility is demonstrated in the afore-described nutritional
epidemiology applications in estimating covariate-adjusted long term intakes of
different dietary components. Supplementary materials include substantive
additional details and R codes are also available online.
