Inference and Interpretability in Latent Variable Modeling

In this age of technology, more and more data is generated as an outcome of complex processes through heterogeneous mechanisms. Statistical models therefore need to invoke the complexities for appropriate inference. One such situation is when data is generated from heterogeneous sub-populations. Hierarchical models form the state-of-the-art methods for such scenarios. However, Markov Chain Monte Carlo methods, which form the traditional inference method can be quite cumbersome to implement for such large scale models, resulting in high time complexities. Sometimes, they may also suffer from inconsistency issues. On the other hand, mean-field variational inference methods even though fast, can suffer from inaccuracy in estimation. This dissertation focuses on understanding such complex models and drawing inference from each of those sub-populations and develops alternative techniques for inference, with statistical guarantees. Our specific contributions are as follows. We provide an in-depth analysis of two specific types of latent variable models, namely, mixture and admixture models and also develop an unsupervised learning scheme with applications to autonomous vehicles. In the mixture model context, firstly, we develop an understanding of the posterior contraction behavior of parameter estimation in the case of infinite mixture models, corresponding to two choices of priors, one parametric and the other nonparametric. Next, we provide an in-depth analysis of Bayesian mixtures under various misspecification settings and provide an asymptotic characterization pertaining to such scenarios. Our study reveals a deep perception of the role, the kernel plays on the statistical decisions of a practitioner. Next, in the context of admixture models, we develop a geometric estimation mechanism to the well-known Latent Dirichlet Allocation model. Finally, we provide a model-free inference scheme to robustly estimate and evaluate parameters of various sub-populations, in the applied setting when the heterogeneous sub-populations for data generation are derived from car driving scenarios, via the use of unsupervised learning.PhDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163255/1/aritra_1.pd