State and Parameter Estimation of HMMs with Transition Density Function
|School||National University of Defense Science and Technology|
|Course||Probability Theory and Mathematical Statistics|
|Keywords||HMM Transition Kernel Importance Sampling SMC Particle Filters MCMC|
HMM(Hidden Markov Models), which were brought forward by Baum and others in the late sixties of the twentieth century, are the most successful statistical modeling ideas that have came up in the last forty years. It has been widely used in many different areas such as speech recognition, anomaly detection and computational biology.Theoretically speaking, HMM need to address three issues: identification problems, hidden state estimation and parameter estimation problems. They are issues form the theoretical basis of HMM, and are often inseparable in practice. This dissertation focuses on hidden state estimation algorithms and parameter estimation algorithms which are universal to all kinds of HMM. We treat the hidden state estimation and parameter estimation problems as optimization problems, and start from the deterministic algorithms and Monte Carlo algorithms to search for answers.Firstly, We review the status of HMM in brief and then define a HMM which can contain most applications of HMM using transition kernel, called HMM with transition kernel, if its transition kernel has a density function, we treat it as HMM with transition density function. All discussion of this paper will be devoted to HMM with transition density function.For hidden state estimation problems, we give Algorithm I based on the principle of single-point optimal and Viterbi algorithm based on the principle of path optimal. On the other hand, we regard it as Bayes filtering problems, and particle filters based on Monte Carlo simulation will be used to obtain approximations of its Bayes Solution.For parameter estimation problems, we first give Baum-Welch algorithm based on the Maximum Likelihood Method and EM algorithm, and then give algorithm of parameter estimation of HMM via Gibbs Sampling based on MCMC (Markov Chain Monte Carlo).