  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, 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.
