This report presents a series of studies related to Hidden Markov Model (HMM), focusing on three major methodological directions: scalable parameter estimation, flexible state sequence recognition, and the learning performance of classification models. First, to address the challenge of high-dimensional hidden states and large-scale datasets, we propose a maximum entropy estimation method based on two-dimensional empirical distributions, which substantially reduces the computational complexity of traditional likelihood-based estimation methods such as the Baum–Welch EM algorithm. Second, to overcome the rigidity of the Viterbi algorithm in state identification, we develop a dynamic programming decoding algorithm grounded in the Markov loss function, allowing more flexible handling of different transition structures through adjustable loss weighting. Furthermore, to enhance the learning performance of classification models, we establish generalization error bounds for Markov sampling and propose a unified ueMC framework that supports both linear and nonlinear models, enabling efficient sampling and training in complex data environments.