This textbook gives a systematic treatment of statistical machine learning for astronomical research, based on Bayesian inference. It shows how modern data analysis techniques connect to classical statistical methods and follow from familiar statistical principles. The Bayesian perspective, used throughout, gives priority to uncertainty quantification and statistical rigor, which scientific inference in astronomy needs. The textbook progresses from probability theory and Bayesian inference through supervised learning, including linear regression with measurement uncertainties, logistic regression and classification. Unsupervised learning topics cover Principal Component Analysis and clustering methods. We then introduce sampling and Markov Chain Monte Carlo, followed by Gaussian Processes as probabilistic nonparametric methods and neural networks in the broader statistical context. Our theory-focused approach derives each method from first principles with complete mathematical development and then shows its astronomical applications. We put statistical insight first and explain why algorithms work, when they are appropriate, and how they connect to broader statistical principles. From a solid grounding in classical methods and their theory, the treatment leads to modern techniques such as neural networks. With this grounding, readers can apply the methods to astronomical research with proper attention to assumptions, limits and uncertainty propagation, as large astronomical surveys require. The 22 companion tutorials apply the methods to astronomical data in Jupyter notebooks.

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