Application of Stochastic Processes in Nonparametric Bayes
This thesis presents theoretical studies of some stochastic processes and their appli- cations in the Bayesian nonparametric methods. The stochastic processes discussed in the thesis are mainly the ones with independent increments - the Levy processes. We develop new representations for the Levy measures of two representative exam- ples of the Levy processes, the beta and gamma processes. These representations are manifested in terms of an infinite sum of well-behaved (proper) beta and gamma dis- tributions, with the truncation and posterior analyses provided. The decompositions provide new insights into the beta and gamma processes (and their generalizations), and we demonstrate how the proposed representation unifies some properties of the two, as these are of increasing importance in machine learning.
Next a new Levy process is proposed for an uncountable collection of covariate- dependent feature-learning measures; the process is called the kernel beta process. Available covariates are handled efficiently via the kernel construction, with covari- ates assumed observed with each data sample ("customer"), and latent covariates learned for each feature ("dish"). The dependencies among the data are represented with the covariate-parameterized kernel function. The beta process is recovered as a limiting case of the kernel beta process. An efficient Gibbs sampler is developed for computations, and state-of-the-art results are presented for image processing and music analysis tasks.
Last is a non-Levy process example of the multiplicative gamma process applied in the low-rank representation of tensors. The multiplicative gamma process is applied along the super-diagonal of tensors in the rank decomposition, with its shrinkage property nonparametrically learns the rank from the multiway data. This model is constructed as conjugate for the continuous multiway data case. For the non- conjugate binary multiway data, the Polya-Gamma auxiliary variable is sampled to elicit closed-form Gibbs sampling updates. This rank decomposition of tensors driven by the multiplicative gamma process yields state-of-art performance on various synthetic and benchmark real-world datasets, with desirable model scalability.
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