Hidden Markov Models (HMMs) are fundamental tools for modeling sequential data, yet traditional methods for evaluation and estimation often struggle with computational inefficiency, especially when applied to large datasets. In such cases, the evaluation of sequence likelihoods becomes imprecise, impairing the performance of the Baum-Welch algorithm, which relies on these likelihood calculations to propose transition and emission matrices. In this work, we present two innovative techniques aimed at improving computational efficiency in these processes. For model evaluation, we introduce a non-parametric, sample-based method that can be used to compare likelihoods using small, randomly sampled segments of data, thereby reducing computational load while preserving statistical integrity. For parameter estimation, we leverage a novel mathematical relationship between transition and emission matrices and observed transitions, which constrains the search space and eliminates the need for iterative likelihood computations. Results from Monte Carlo simulations demonstrate that our methods substantially enhance computational performance without compromising accuracy.