Fundamental Tradeoffs in Estimation of Finite-State Hidden Markov Models
Justin H. Le · Digital Scholarship - UNLV (University of Nevada Reno) · 2020
Hidden Markov models (HMMs) constitute a broad and flexible class of statistical models that are widely used in studying processes that evolve over time and are only observable through the collection of noisy data. Two problems are essential to the use of HMMs: state estimation and parameter estimation. In state estimation, an algorithm estimates the sequence of states of the process that most likely generated a certain sequence of observations in the data. In parameter estimation, an algorithm computes the probability distributions that govern the time-evolution of states and the sampling of data. Although algorithms for the two problems are widely researched, relatively little study has been devoted to understanding the tradeoffs between key design variables of these algorithms from a mathematically rigorous viewpoint. In this thesis, we provide such a study by establishing theorems regarding these tradeoffs. Furthermore, we illustrate the implications of these theorems in practice, highlighting the scope of their applicability and generality. We then suggest directions for future research in this area by bringing attention to the critical assumptions and tools used in the proofs of our theorems.