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Scientific studies in many fields involve understanding and characterizing dependence relationships among large numbers of variables. This can be challenging in settings where data is limited and noisy. Take survey data as an example, understanding the associations between questions may help researchers better explain themes amongst related questions and impute missing values. Yet, such data typically contains a combination of binary, continuous, and categorical variables; a high proportion of missing values; and complex data structures. In this talk, we develop flexible models and algorithms to estimate Gaussian and latent Gaussian graphical models from noisy data. First, we develop a latent Gaussian graphical model for mixed data that takes advantage of informative prior beliefs on the marginal distribution of variables. Next, we propose several shrinkage priors for precision matrices and develop estimation procedures for fast posterior explorations of single and multiple graphical models. This work is motivated by modeling survey-based cause of death instruments, known as verbal autopsies (VAs). Our methods can be integrated into existing probabilistic cause-of-death assignment algorithms and improve model performance while recovering useful dependencies in the data.