Inferential Models: Reasoning with Uncertainty, 1st Edition (Hardback) book cover

Inferential Models

Reasoning with Uncertainty, 1st Edition

By Ryan Martin, Chuanhai Liu

Chapman and Hall/CRC

256 pages | 37 B/W Illus.

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pub: 2015-09-25
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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.


"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

Table of Contents



Assumed background

Scientific inference: An overview

Prediction and inference

Outline of the book

Prior-Free Probabilistic Inference


Probabilistic inference

Existing approaches

On the role of probability in statistical inference

Our contribution in this book

Two Fundamental Principles


Validity principle

Efficiency principle


On conditioning for improved efficiency

Inferential Models


Basic overview

Inferential models

Theoretical validity of IMs

Theoretical optimality of IMs

Two more examples

Concluding remarks

Predictive Random Sets


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


Conditional IMs

Finding conditional associations

Three detailed examples

Local conditional IMs

Concluding remarks

Marginal Inferential Models


Marginal inferential models


Marginal IMs for non-regular models

Concluding remarks

Normal Linear Models


Linear regression

Linear mixed effect models

Concluding remarks

Prediction of Future Observations


Inferential models for prediction

Examples and applications

Some further technical details

Concluding remarks

Simultaneous Inference on Multiple Assertions



Classification of assertions

Optimality for a collection of assertions

Optimal IMs for variable selection

Concluding remarks

Generalized Inferential Models


Generalized associations

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

Final remarks



Exercises appear at the end of each chapter.

About the Authors

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.

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General