Conditional Measures and Applications: 2nd Edition (Hardback) book cover

Conditional Measures and Applications

2nd Edition

By M.M. Rao

Chapman and Hall/CRC

506 pages

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pub: 2005-05-25
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Description

In response to unanswered difficulties in the generalized case of conditional expectation and to treat the topic in a well-deservedly thorough manner, M.M. Rao gave us the highly successful first edition of Conditional Measures and Applications. Until this groundbreaking work, conditional probability was relegated to scattered journal articles and mere chapters in larger works on probability. This second edition continues to offer a thorough treatment of conditioning while adding substantial new information on developments and applications that have emerged over the past decade.

Conditional Measures and Applications, Second Edition clearly elucidates the subject, from fundamental principles to abstract analysis. The author illustrates the computational difficulties in evaluating conditional probabilities in nondiscrete cases with numerous examples, demonstrates applications to Markov processes, martingales, potential theory, and Reynolds operators as well as sufficiency in statistics, and clarifies ideas in modern noncommutative probability structures through conditioning in general structures, including parts of operator algebras and "free" random variables. He also discusses existence and construction problems from the Bishop-Brouwer constructive analysis point of view.

With open problems in every chapter and links to other areas of mathematics, this invaluable second edition offers complete coverage of conditional probability and expectation and their structural analysis, from simple to advanced abstract levels, for both novices and seasoned mathematicians.

Table of Contents

THE CONCEPT OF CONDITIONING

Introduction

Conditional Probability Given a Partition

Conditional Expectation: Elementary Case

Conditioning with Densities

Conditional Probability Spaces: First Steps

Bibliographical Notes

THE KOLMOGOROV FORMULATION AND ITS PROPERTIES

Introduction of the General Concept

Basic Properties of Conditional Expectations

Conditional Probabilities in the General Case

Remarks on the Inclusion of Previous Concepts

Conditional Independence and Related Concepts

Bibliographical Notes

COMPUTATIONAL PROBLEMS ASSOCIATED WITH CONDITIONING

Introduction

Some Examples with Multiple Solutions: Paradoxes

Dissection of Paradoxes

Some Methods of Computation

Remarks on Traditional Calculations of Conditional Measures

Bibliographical Notes

AN AXIOMATIC APPROACH TO CONDITIONAL PROBABILITY

Introduction

Axiomatization of Conditioning Based on Partitions

Structure of the New Conditional Probability Functions

Some Applications

Difficulties with Earlier Examples Persist

Bibliographical Notes

REGULARITY OF CONDITIONAL MEASURES

Introduction

Existence of Regular Conditional Probabilities: Generalities

Special Spaces Admitting Regular Conditional Probabilities

Disintegration of Probability Measures and Regular Conditioning

Further Results on Disintegration

Evaluation of Conditional Expectations by Fourier Analysis

Further Evaluations of Conditional Expectations

Bibliographical Notes

SUFFICIENCY

Introduction

Conditioning Relative to Families of Measures

Sufficiency: The Dominated Case

Sufficiency: The Undominated Case

Sufficiency: Another Approach to the Undominated Case

Bibliographical Notes

ABSTRACTION OF KOLMOGOROV'S FORMULATION

Introduction

Integration Relative to Conditional Measures and Function Spaces

Functional Characterizations of Conditioning

Integral Representations of Conditional Expectations

Rényi's Formulation as a Specialization of the Abstract Version

Conditional Measures and Differentiation

Bibliographical Notes

PRODUCTS OF CONDITIONAL MEASURES

Introduction

A General Formulation of Products

General Projective Limit Theorems

Some Consequences

Remarks on Conditioning, Disintegration, and Lifting

Bibliographical Notes

APPLICATIONS TO MARTINGALES AND MARKOV PROCESSES

Introduction

Set Martingales

Martingale Convergence

Markov Processes: Some Basic Results

Further Properties of Markov Processes

Bibliographical Notes

APPLICATIONS TO MODERN ANALYSIS

Introduction and Motivation

Conditional Measures and Potential Kernels

Reynolds Operators and Conditional Expectations

Bistochastic Operators and Conditioning

Contractive Projections and Conditional Expectations

Bibliographical Notes

CONDITIONING IN GENERAL STRUCTURES

Introduction

Averagings in Cones of Positive Functions

Averaging Operators on Function Algebras

Conditioning in Operator Algebras

Free Independence and a Bijection in Operator Algebras

Some Applications of Noncommutative Conditioning

Bibliographical Notes

REFERENCES

NOTATIONS

AUTHOR INDEX

SUBJECT INDEX

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis
SCI040000
SCIENCE / Mathematical Physics