Conditional Measures and Applications SECOND EDITION

The second edition of this authoritative text offers an in-depth treatment of all aspects of conditional expectations and probability measures and their structural analysis. However, what makes this the definitive reference on conditional measures is the author's keen ability to link theory with practical applications. The author covers applications in sufficiency, Markov processes and martingales. He also illustrates the difficulties in calculating conditional expectations for continuous multivariate distributions and methods for correct solutions in such cases. Each chapter concludes with several open-ended problems that reinforce principles and promote practical problem solving.

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.

THE CONCEPT OF CONDITIONINGIntroductionConditional Probability Given a PartitionConditional Expectation: Elementary CaseConditioning with DensitiesConditional Probability Spaces: First StepsBibliographical NotesTHE KOLMOGOROV FORMULATION AND ITS PROPERTIESIntroduction of the General ConceptBasic Properties of Conditional ExpectationsConditional Probabilities in the General CaseRemarks on the Inclusion of Previous ConceptsConditional Independence and Related ConceptsBibliographical NotesCOMPUTATIONAL PROBLEMS ASSOCIATED WITH CONDITIONINGIntroductionSome Examples with Multiple Solutions: ParadoxesDissection of ParadoxesSome Methods of ComputationRemarks on Traditional Calculations of Conditional MeasuresBibliographical NotesAN AXIOMATIC APPROACH TO CONDITIONAL PROBABILITYIntroductionAxiomatization of Conditioning Based on PartitionsStructure of the New Conditional Probability FunctionsSome ApplicationsDifficulties with Earlier Examples PersistBibliographical NotesREGULARITY OF CONDITIONAL MEASURESIntroductionExistence of Regular Conditional Probabilities: GeneralitiesSpecial Spaces Admitting Regular Conditional ProbabilitiesDisintegration of Probability Measures and Regular ConditioningFurther Results on DisintegrationEvaluation of Conditional Expectations by Fourier AnalysisFurther Evaluations of Conditional ExpectationsBibliographical NotesSUFFICIENCYIntroductionConditioning Relative to Families of MeasuresSufficiency: The Dominated CaseSufficiency: The Undominated CaseSufficiency: Another Approach to the Undominated CaseBibliographical NotesABSTRACTION OF KOLMOGOROV'S FORMULATIONIntroductionIntegration Relative to Conditional Measures and Function SpacesFunctional Characterizations of ConditioningIntegral Representations of Conditional ExpectationsRényi's Formulation as a Specialization of the Abstract VersionConditional Measures and DifferentiationBibliographical NotesPRODUCTS OF CONDITIONAL MEASURESIntroductionA General Formulation of ProductsGeneral Projective Limit TheoremsSome ConsequencesRemarks on Conditioning, Disintegration, and LiftingBibliographical NotesAPPLICATIONS TO MARTINGALES AND MARKOV PROCESSESIntroductionSet MartingalesMartingale ConvergenceMarkov Processes: Some Basic ResultsFurther Properties of Markov ProcessesBibliographical NotesAPPLICATIONS TO MODERN ANALYSISIntroduction and MotivationConditional Measures and Potential KernelsReynolds Operators and Conditional ExpectationsBistochastic Operators and ConditioningContractive Projections and Conditional ExpectationsBibliographical NotesCONDITIONING IN GENERAL STRUCTURESIntroductionAveragings in Cones of Positive FunctionsAveraging Operators on Function AlgebrasConditioning in Operator AlgebrasFree Independence and a Bijection in Operator AlgebrasSome Applications of Noncommutative ConditioningBibliographical NotesREFERENCESNOTATIONSAUTHOR INDEXSUBJECT INDEX