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Explore the evolution and future directions of nonparametric statistical inference when data must satisfy specific shape constraints in this comprehensive lecture delivered at the International Conference on Bayesian Statistics 2025. Delve into the mathematical foundations and practical applications of shape-constrained estimation methods, examining how researchers can make statistical inferences about unknown functions while incorporating prior knowledge about their geometric properties such as monotonicity, convexity, or unimodality. Learn about the historical development of this field, current state-of-the-art methodologies, and emerging research directions that promise to advance nonparametric statistics. Discover how shape constraints can improve estimation accuracy, provide more interpretable results, and enable robust inference in various applications across statistics, machine learning, and data science. Gain insights into computational algorithms, theoretical guarantees, and practical considerations for implementing shape-constrained methods in real-world scenarios.
