Bayesian Methods, Computation, Model Selection and Goodness of Fit with Complex Data

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My primary research interests lie broadly in Bayesian methods and computation for complex models which integrate both modelling and computational aspects. In my research program, I develop, study, and implement methods for providing statistical inference for large or complicated data arising from health, environmental, sports and social sciences. In this research proposal, I'm particularly interested in developing novel Bayesian methods for complex data types arising from social networks, environmental, ecological and health systems.We are in the era of data science where enormous amounts of data arise in every discipline with various complexities. These complexities could be in the form of incompleteness, dimensionality, complex structures or big data. In developing a statistical model for these data, we typically embody a set of statistical assumptions concerning the generation of data, either uncertain future data or already observed data. This involves linking a set of parameters to the data in a structural fashion. This can be achieved using a parametric or nonparametric model. In a parametric model, the parameters are in finite-dimensional spaces while the parameters are in infinite-dimensional spaces in a nonparametric model. Note that the parameters are assumed to be fixed unknowns in both approaches. In Bayesian models, we assume that these parameters arise from their own distributions called prior distributions. A prior distribution is used to quantify our prior knowledge about the parameters. When data are available, we can update our prior knowledge using the conditional distribution of parameters, given the data. I will develop generative Bayesian hierarchical models where parameter spaces lie in finite dimensions. I will then consider nonparametric Bayesian models whose parameter space has infinite dimension. To define a nonparametric Bayesian model, we have to define a probability distribution on an infinite dimensional space. These models are useful to model distributions as mixtures of simpler distributions and to identify latent classes that can explain the complex dependencies between variables. This allows using a countably infinite number of mixtures, which bypasses the need to determine the correct number of components in a finite mixture model. When models are based on conjugate prior distributions, sampling from the posterior distribution of the parameters of the component distributions and/or of the associations of mixture components is feasible through Gibbs sampling. When they are not conjugate, I will develop Markov chain Monte Carlo (MCMC) sampling methods to learn about the model parameters. Model selection and goodness-of-fit methods will also be developed. The methods developed here will be useful to predict the online social network structures, newborn safety and child health, environmental behavior in Hudson Bay and Eastern Canadian Arctic.