A New Approach to Sound Statistical Reasoning Inferential Models: Reasoning with Uncertainty introduces the authors' recently developed approach to inference: the inferential model (IM) framework. This logical framework for exact probabilistic inference does not require the user to input prior information. The authors show how an IM produces meaningful prior-free probabilistic inference at a high level. The book covers the foundational motivations for this new IM approach, the basic theory behind its calibration properties, a number of important applications, and new directions for research. It discusses alternative, meaningful probabilistic interpretations of some common inferential summaries, such as p-values. It also constructs posterior probabilistic inferential summaries without a prior and Bayes' formula and offers insight on the interesting and challenging problems of conditional and marginal inference. This book delves into statistical inference at a foundational level, addressing what the goals of statistical inference should be. It explores a new way of thinking compared to existing schools of thought on statistical inference and encourages you to think carefully about the correct approach to scientific inference.
Preliminaries Introduction Assumed background Scientific inference: An overview Prediction and inference Outline of the book Prior-Free Probabilistic Inference Introduction Probabilistic inference Existing approaches On the role of probability in statistical inference Our contribution in this book Two Fundamental Principles Introduction Validity principle Efficiency principle Recap On conditioning for improved efficiency Inferential Models Introduction Basic overview Inferential models Theoretical validity of IMs Theoretical optimality of IMs Two more examples Concluding remarks Predictive Random Sets Introduction Random sets Predictive random sets for constrained problems Theoretical results on elastic predictive random sets Two examples of the elastic belief method Concluding remarks Conditional Inferential Models Introduction Conditional IMs Finding conditional associations Three detailed examples Local conditional IMs Concluding remarks Marginal Inferential Models Introduction Marginal inferential models Examples Marginal IMs for non-regular models Concluding remarks Normal Linear Models Introduction Linear regression Linear mixed effect models Concluding remarks Prediction of Future Observations Introduction Inferential models for prediction Examples and applications Some further technical details Concluding remarks Simultaneous Inference on Multiple Assertions Introduction Preliminaries Classification of assertions Optimality for a collection of assertions Optimal IMs for variable selection Concluding remarks Generalized Inferential Models Introduction Generalized associations A generalized IM A generalized marginal IM Remarks on generalized IMs Application: Large-scale multinomial inference Future Research Topics Introduction New directions to explore Our "top ten list" of open problems Final remarks Bibliography Index Exercises appear at the end of each chapter.
