Abstract:
<p>Bayesian nonparametric methods are useful for modeling data without having to define the complexity of the entire model <italic>a priori</italic>, but rather allowing for this complexity to be determined by the data. Two problems considered in this dissertation are the number of components in a mixture model, and the number of factors in a latent factor model, for which the Dirichlet process and the beta process are the two respective Bayesian nonparametric priors selected for handling these issues.</p>
<p>The flexibility of Bayesian nonparametric priors arises from the prior's definition over an infinite dimensional parameter space. Therefore, there are theoretically an <italic>infinite</italic> number of latent components and an <italic>infinite</italic> number of latent factors. Nevertheless, draws from each respective prior will produce only a small number of components or factors that appear in a given data set. As mentioned, the number of these components and factors, and their corresponding parameter values, are left for the data to decide.</p>
<p>This dissertation is split between novel practical applications and novel theoretical results for these priors. For the Dirichlet process, we investigate stick-breaking representations for the finite Dirichlet process and their application to novel sampling techniques, as well as a novel mixture modeling framework that incorporates multiple modalities within a data set. For the beta process, we present a new stick-breaking construction for the infinite-dimensional prior, and consider applications to image interpolation problems and dictionary learning for compressive sensing.</p>
