Principles of Uncertainty

Principles of Uncertainty

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Features

2011 Degroot Prize Winner!

  • Well-written and comprehensive introduction to theory of Bayesian statistics
  • Each of the introductory chapters begins by introducing one new concept or assumption
  • Uses "just-in-time mathematics"—the introduction to mathematical ideas just before they are applied
  • Each chapter ends with a summary and exercises

Summary

An intuitive and mathematical introduction to subjective probability and Bayesian statistics.

An accessible, comprehensive guide to the theory of Bayesian statistics, Principles of Uncertainty presents the subjective Bayesian approach, which has played a pivotal role in game theory, economics, and the recent boom in Markov Chain Monte Carlo methods. Both rigorous and friendly, the book contains:

  • Introductory chapters examining each new concept or assumption
  • Just-in-time mathematics – the presentation of ideas just before they are applied
  • Summary and exercises at the end of each chapter
  • Discussion of maximization of expected utility
  • The basics of Markov Chain Monte Carlo computing techniques
  • Problems involving more than one decision-maker

Written in an appealing, inviting style, and packed with interesting examples, Principles of Uncertainty introduces the most compelling parts of mathematics, computing, and philosophy as they bear on statistics. Although many books present the computation of a variety of statistics and algorithms while barely skimming the philosophical ramifications of subjective probability, this book takes a different tack. By addressing how to think about uncertainty, this book gives readers the intuition and understanding required to choose a particular method for a particular purpose.

Table of Contents

Probability
Avoiding being a sure loser
Disjoint events
Events not necessarily disjoint
Random variables, also known as uncertain quantities
Finite number of values
Other properties of expectation
Coherence implies not a sure loser
Expectations and limits

Conditional Probability and Bayes Theorem
Conditional probability
The Birthday Problem
Simpson's Paradox
Bayes Theorem
Independence of events
The Monty Hall problem
Gambler's Ruin problem
Iterated Expectations and Independence
The binomial and multinomial distributions
Sampling without replacement
Variance and covariance
A short introduction to multivariate thinking
Tchebychev's inequality

Discrete Random Variables
Countably many possible values
Finite additivity
Countable Additivity
Properties of countable additivity
Dynamic sure loss

Probability generating functions
Geometric random variables
The negative binomial random variable
The Poisson random variable
Cumulative distribution function
Dominated and bounded convergence

Continuous Random Variables
Introduction
Joint distributions
Conditional distributions and independence
Existence and properties of expectations
Extensions
An interesting relationship between cdf's and expectations of continuous random variables
Chapter retrospective so far
Bounded and dominated convergence
The Riemann-Stieltjes integral
The McShane-Stieltjes Integral
The road from here
The strong law of large numbers

Transformations
Introduction
Discrete Random Variables
Univariate Continuous Distributions
Linear spaces
Permutations
Number systems; DeMoivre's formula
Determinants
Eigenvalues, eigenvectors and decompositions
Non-linear transformations
The Borel-Kolmogorov paradox

Normal Distribution
Introduction
Moment generating functions
Characteristic functions
Trigonometric Polynomials
A Weierstrass approximation theorem
Uniqueness of characteristic functions
Characteristic function and moments
Continuity Theorem
The Normal distribution
Multivariate normal distributions
Limit theorems

Making Decisions
Introduction
An example
In greater generality
The St. Petersburg Paradox
Risk aversion
Log (fortune) as utility
Decisions after seeing data
The expected value of sample information
An example
Randomized decisions
Sequential decisions

Conjugate Analysis
A simple normal-normal case
A multivariate normal case, known precision
The normal linear model with known precision
The gamma distribution
Uncertain Mean and Precision
The normal linear model, uncertain precision
The Wishart distribution
Both mean and precision matrix uncertain
The beta and Dirichlet distributions
The exponential family
Large sample theory for Bayesians
Some general perspective

Hierarchical Structuring of a Model
Introduction
Missing data
Meta-analysis
Model uncertainty/model choice
Graphical Hierarchical Models
Causation

Markov Chain Monte Carlo
Introduction
Simulation
The Metropolis Hasting Algorithm
Extensions and special cases
Practical considerations
Variable dimensions: Reversible jumps

Multiparty Problems
A simple three-stage game
Private information
Design for another's analysis
Optimal Bayesian Randomization
Simultaneous moves
The Allais and Ellsberg paradoxes
Forming a Bayesian group

Exploration of Old Ideas
Introduction
Testing
Confidence intervals and sets
Estimation
Choosing among models
Goodness of fit
Sampling theory statistics
Objective" Bayesian Methods

Epilogue: Applications
Computation
A final thought

Author Bio(s)

Joseph B. Kadane, Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA

Editorial Reviews

This book mainly focuses on the use of Bayesian statistics. It is written using stories and many examples to which readers can relate, and is thus an engaging and appealing text on what is generally a very dry mathematical subject.
—John J. Shea, IEEE Electrical Insulation Magazine, May/June 2013, Vol. 29, No. 3

This text provides a unique blend of theory, methods, philosophy and applications that is suitable for a course in Bayesian probability and statistics. … provides thought-provoking material for teaching. …
—Erkki P. Liski, International Statistical Review, 2012

In this remarkable book, Kadane begins at the most rudimentary level, develops all the needed mathematics on the fly, and still manages to flesh out at least the core of the whole story, slowly, thoughtfully, and rigorously, right up to graduate level. Major theorems all proved in detail appear here, but not for their own sake; the author always carefully selects them to clarify the basic meaning of the subject and his own views concerning the pitfalls and subtleties of its proper application. Summing Up: Highly recommended.
—D.V. Feldman, CHOICE, February 2012

Principles of Uncertainty is a profound and mesmerising book on the foundations and principles of subjectivist or behaviouristic Bayesian analysis. … the book is a pleasure to read. And highly recommended for teaching as it can be used at many different levels. … A must-read for sure!
—Christian Robert, The Statistics Forum/CHANCE, October 2011

It's a lovely book, one that I hope will be widely adopted as a course textbook.
—Michael Jordan, University of California, Berkeley, USA

A careful, complete, and lovingly written exposition of the subjective Bayesian viewpoint by one of its most eloquent and staunch defenders. Summarizes a lifetime of theory, methods, and application developments for the Bayesian inferential engine. A must-read for anyone looking for a deep understanding of the foundations of Bayesian methods and what they offer modern statistical practice.
—Bradley P. Carlin, Professor and Head of Division of Biostatistics, University of Minnesota, Minneapolis, USA

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