Statistical information

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This monograph builds on the foundational work of Arthur Dempster, Glenn Shafer, and R. A. Fisher in statistical inference, focusing on the concept of statistical information. It introduces a novel inferential mechanism termed assumption-based reasoning, which blends logic and classical probability theory. While traces of this idea appear in Jacob Bernoulli's writings, the book offers a comprehensive description, showcasing that assumption-based inference on functional models generalizes both Bayesian and Fisher's fiducial inference, addressing the longstanding debate between these theories. The approach clarifies post-data probabilistic statements about unknown parameters, revealing that the likelihood function does not encompass all statistical information from an experiment. The book describes statistical information through functional models, illustrating how observations relate to unknown parameters and stochastic disturbances. The first part examines discrete functional models, presenting assumption-based reasoning without technical complexities, demonstrating how to combine information to focus on specific questions. It introduces an algebraic perspective on statistical information analysis and preliminary results on hypothesis selection. The second part explores continuous models through assumption-based reasoning, clarifying concepts in Fisher's fiducial theory and the role of improper priors in Bayesian inference. Th