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Inferential Models
Reasoning with Uncertainty




ISBN 9781439886489
Published September 25, 2015 by Chapman and Hall/CRC
276 Pages 37 B/W Illustrations

 
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Book Description

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

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.

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Author(s)

Biography

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.

Reviews

"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