Principles of Uncertainty

1. Auflage 2011
ISBN: 978-1-4398-6161-5
Verlag: Chapman and Hall/CRC

Buch, Englisch, 504 Seiten, Format (B × H): 187 mm x 277 mm, Gewicht: 1099 g

Reihe: Chapman & Hall/CRC Texts in Statistical Science

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.

Zielgruppe

Graduate students and researchers in statistics.

Autoren/Hrsg.

Kadane, Joseph B.

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik
• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

ProbabilityAvoiding being a sure loserDisjoint 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 TheoremConditional 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 VariablesCountably many possible valuesFinite additivityCountable AdditivityProperties of countable additivityDynamic sure lossProbability generating functionsGeometric random variablesThe negative binomial random variableThe Poisson random variableCumulative distribution functionDominated and bounded convergenceContinuous Random VariablesIntroductionJoint distributionsConditional distributions and independenceExistence and properties of expectationsExtensionsAn interesting relationship between cdf's and expectations of continuous random variablesChapter retrospective so farBounded and dominated convergenceThe Riemann-Stieltjes integralThe McShane-Stieltjes IntegralThe road from hereThe strong law of large numbersTransformationsIntroductionDiscrete Random VariablesUnivariate Continuous DistributionsLinear spacesPermutationsNumber systems; DeMoivre's formulaDeterminantsEigenvalues, eigenvectors and decompositionsNon-linear transformationsThe Borel-Kolmogorov paradoxNormal DistributionIntroductionMoment generating functionsCharacteristic functionsTrigonometric PolynomialsA Weierstrass approximation theoremUniqueness of characteristic functionsCharacteristic function and momentsContinuity TheoremThe Normal distributionMultivariate normal distributionsLimit theoremsMaking DecisionsIntroductionAn exampleIn greater generalityThe St. Petersburg ParadoxRisk aversionLog (fortune) as utilityDecisions after seeing dataThe expected value of sample informationAn exampleRandomized decisionsSequential decisionsConjugate AnalysisA simple normal-normal caseA multivariate normal case, known precisionThe normal linear model with known precisionThe gamma distributionUncertain Mean and PrecisionThe normal linear model, uncertain precisionThe Wishart distributionBoth mean and precision matrix uncertainThe beta and Dirichlet distributionsThe exponential familyLarge sample theory for BayesiansSome general perspectiveHierarchical Structuring of a ModelIntroductionMissing dataMeta-analysisModel uncertainty/model choiceGraphical Hierarchical ModelsCausationMarkov Chain Monte CarloIntroductionSimulationThe Metropolis Hasting AlgorithmExtensions and special casesPractical considerationsVariable dimensions: Reversible jumpsMultiparty ProblemsA simple three-stage gamePrivate informationDesign for another's analysisOptimal Bayesian RandomizationSimultaneous movesThe Allais and Ellsberg paradoxesForming a Bayesian groupExploration of Old IdeasIntroductionTestingConfidence intervals and setsEstimationChoosing among modelsGoodness of fitSampling theory statisticsObjective" Bayesian MethodsEpilogue: ApplicationsComputationA final thought

Über Autor(innen)

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