We discuss divergence measures and applications encompassing some areas of machine learning, Boltzmann machines, gradient boosting, active learning, and cosine similarity. Boltzmann machines have wide developments for generative models by the help of statistical dynamics. The ML-estimator is a basic device for data learning, but the computation is challenging for evaluating the partition functions. We introduce the GM-divergence and the GM-estimator for the Boltzmann machines. The GM-estimator is shown a fast computation thanks to free evaluation of the partition function. Next, we focus on active learning, particularly the Query by Committee method. It highlights how divergence measures can be used to select informative data points, integrating statistical and machine learning concepts. Finally, we extend the \(\gamma \) -divergence on a space of real-valued functions. This yields a natural extension of the cosine similarities, called \(\gamma \) -cosine similarities. The basic properties are explored and demonstrated in numerical experiments compared to traditional cosine similarity.