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.
Table of Contents
Scientific inference: An overview
Prediction and inference
Outline of the book
Prior-Free Probabilistic Inference
On the role of probability in statistical inference
Our contribution in this book
Two Fundamental Principles
On conditioning for improved efficiency
Theoretical validity of IMs
Theoretical optimality of IMs
Two more examples
Predictive Random Sets
Predictive random sets for constrained problems
Theoretical results on elastic predictive random sets
Two examples of the elastic belief method
Conditional Inferential Models
Finding conditional associations
Three detailed examples
Local conditional IMs
Marginal Inferential Models
Marginal inferential models
Marginal IMs for non-regular models
Normal Linear Models
Linear mixed effect models
Prediction of Future Observations
Inferential models for prediction
Examples and applications
Some further technical details
Simultaneous Inference on Multiple Assertions
Classification of assertions
Optimality for a collection of assertions
Optimal IMs for variable selection
Generalized Inferential Models
A generalized IM
A generalized marginal IM
Remarks on generalized IMs
Application: Large-scale multinomial inference
Future Research Topics
New directions to explore
Our "top ten list" of open problems
Exercises appear at the end of each chapter.
Ryan Martin is an associate professor in the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago.
Chuanhai Liu is a professor in the Department of Statistics at Purdue University.
"The book . . . delivers on its promise. It should be read by all statisticians with an interest in the foundations and development of the statistical methods for inference."
~Michael J. Lew, University of Melbourne
" . . . the book covers the motivations for the IM framework, the basic theory behind its calibration properties, a number of its applications and gives a new way of thinking compared to existing schools of thought on statistical inference"
~Apostolos Batsidis (Ioannina), Zentralblatt MATH