概率论沉思录

PartⅠ　Principlesandelementaryapplications
1　Plausiblereasoning
1.1　Deductiveandplausiblereasoning
1.2　Analogieswith　slcaltheories
1.3　Thethinkingcomputer
1.4　Introducingtherobot
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PartⅠ　Principlesandelementaryapplications
1　Plausiblereasoning
1.1　Deductiveandplausiblereasoning
1.2　Analogieswith　slcaltheories
1.3　Thethinkingcomputer
1.4　Introducingtherobot
1.5　Booleanalgebra
1.6　Adequatesetsofoperations
1.7　Thebasicdesiderata
1.8　Comments
1.8.1　Commonlanguagevs.formallogic
1.8.2　Nitpicking
2　Thequantitativerules
2.1　Theproductrule
2.2　Thesumrule
2.3　Qualitativeproperties
2.4　Numericalvalues
2.5　Notationandfinite-setspolicy
2.6　Comments
2.6.1　Suectlvevs.oectlve
2.6.2　G/3delstheorem
2.6.3　Venndiagrams
2.6.4　TheKolmogorovaxioms
3　Elementarysamplingtheory
3.1　Samplingwithoutreplacement
3.2　Logicvs.propensity
3.3　Reasoningfromlesspreciseinformation
3.4　Expectations
3.5　Otherformsandextensions
3.6　Probabilityasamathematicaltool
3.7　Thebinomialdistribution
3.8　Samplingwithreplacement
3.8.1　Digression：asermononrealityvs.models
3.9　Correctionforcorrelations
3.10　Simplification
3.11　Comments
3.11.1　Alookahead
4　Elementaryhypothesistesting
4.1　Priorprobabilities
4.2　Testingbinaryhypotheseswithbinarydata
4.3　Nonextensibilitybeyondthebinarycase
4.4　Multiplehypothesistesting
4.4.1　Digressiononanotherderivation
4.5　Continuousprobabilitydistributionfunctions
4.6　Testinganinfinitenumberofhypotheses
4.6.1　Historicaldigression
4.7　Simpleandcompound(orcomposite)hypotheses
4.8　Comments
4.8.1　Etymology
4.8.2　Whathaveweaccomplished?
5　Queerusesforprobabilitytheory
5.1　Extrasensoryperception
5.2　MrsStewartstelepathicpowers
5.2.1　Digressiononthenormalapproximation
5.2.2　BacktoMrsStewart
5.3　Converginganddivergingviews
5.4　Visualperception-evolutionintoBayesianity?
5.5　ThediscoveryofNeptune
5.5.1　Digressiononalternativehypotheses
5.5.2　BacktoNewton
5.6　Horseracingandweatherforecasting
5.6.1　Discussion
5.7　Paradoxesofintuition
5.8　Bayesianjurisprudence
5.9　Comments
5.9.1　Whatisqueer?
6　Elementaryparameterestimation
6.1　Inversionoftheumdistributions
6.2　BothNandRunknown
6.3　Uniformprior
6.4　Predictivedistributions
6.5　Truncateduniformpriors
6.6　Aconcaveprior
6.7　Thebinomialmonkeyprior
6.8　Metamorphosisintocontinuousparameterestimation
6.9　Estimationwithabinomialsamplingdistribution
6.9.1　Digressiononoptionalstopping
6.10　Compoundestimationproblems
6.11　AsimpleBayesianestimate：quantitativepriorinformation
6.11.1　Fromposteriordistributionfunctiontoestimate
6.12　Effectsofqualitativepriorinformation
6.13　Choiceofaprior
6.14　Onwiththecalculation!
6.15　TheJeffreysprior
6.16　Thepointofitall
6.17　Intervalestimation
6.18　Calculationofvariance
6.19　Generalizationandasymptoticforms
6.20　Rectangularsamplingdistribution
6.21　Smallsamples
6.22　Mathematicaltrickery
6.23　Comments
7　Thecentral，Gaussianornormaldistribution
7.1　Thegravitatingphenomenon
7.2　TheHerschel-Maxwellderivation
7.3　TheGaussderivation
7.4　HistoricalimportanceofGausssresult
7.5　TheLandonderivation
7.6　WhytheubiquitoususeofGausslandistributions?
7.7　Whytheubiquitoussuccess?
7.8　Whatestimatorshouldweuse?
7.9　Errorcancellation
7.10　Thenearirrelevanceofsamplingfrequencydistributions
7.11　Theremarkableefficiencyofinformationtransfer
7.12　Othersamplingdistributions
7.13　Nuisanceparametersassafetydevices
7.14　Moregeneralproperties
7.15　ConvolutionofGaussians
7.16　Thecentrallimittheorem
7.17　Accuracyofcomputations
7.18　Galtonsdiscovery
7.19　PopulationdynamicsandDarwinianevolution
7.20　Evolutionofhumming-birdsandflowers
7.21　Applicationtoeconomics
7.22　ThegreatinequalityofJupiterandSaturn
7.23　ResolutionofdistributionsintoGaussians
7.24　Hermitepolynomialsolutions
7.25　Fouriertransformrelations
7.26　Thereishopeafterall
7.27　Comments
7.27.1　Terminologyagain
8　Sufficiency，ancillarity，andallthat
8.1　Sufficiency
8.2　Fishersufficiency
8.2.1　Examples
8.2.2　TheBlackwell-Raotheorem
8.3　Generalizedsufficiency
8.4　Sufficiencyplusnuisanceparameters
8.5　Thelikelihoodprinciple
8.6　Ancillarity
8.7　Generalizedancillaryinformation
8.8　Asymptoticlikelihood：Fisherinformation
8.9　Combiningevidencefromdifferentsources
8.10　Poolingthedata
8.10.1　Fine-grainedpropositions
8.11　Samsbrokenthermometer
8.12　Comments
8.12.1　Thefallacyofsamplere-use
8.12.2　Afolktheorem
8.12.3　Effectofpriorinformation
8.12.4　Clevertricksandgamesmanship
9　Repetitiveexperiments：probabilityandfrequency
9.1　Physicalexperiments
9.2　Thepoorlyinformedrobot
9.3　Induction
9.4　Aretheregeneralinductiverules?
9.5　Multiplicityfactors
9.6　Partitionfunctionalgorithms
9.6.1　Solutionbyinspection
9.7　Entropyalgorithms
9.8　Anotherwayoflookingatit
9.9　Entropymaximization
9.10　Probabilityandfrequency
9.11　Significancetests
9.11.1　Impliedalternatives
9.12　Comparisonofpsiandchi-squared
9.13　Thechi-squaredtest
9.14　Generalization
9.15　Halleysmortalitytable
9.16　Comments
9.16.1　Theirrationalists
9.16.2　Superstitions
10　Physicsofrandomexperiments
10.1　Aninterestingcorrelation
10.2　Historicalbackground
10.3　Howtocheatatcoinanddietossing
10.3.1　Experimentalevidence
10.4　Bridgehands
10.5　Generalrandomexperiments
10.6　Inductionrevisited
10.7　Butwhataboutquantumtheory?
10.8　Mechanicsundertheclouds
10.9　Moreoncoinsandsymmetry
10.10　Independenceoftosses
10.11　Thearroganceoftheuninformed
PartⅡ　Advancedapplications
11　Discretepriorprobabilities：theentropyprinciple
11.1　Anewkindofpriorinformation
11.2　Minimum∑Pi2
11.3　Entropy：Shannonstheorem
11.4　TheWallisderivation
11.5　Anexample
11.6　Generalization：amorerigorousproof
11.7　Formalpropertiesofmaximumentropydistributions
11.8　Conceptualproblems-frequencycorrespondence
11.9　Comments
12　Ignorancepriorsandtransformationgroups
12.1　Whatarewetryingtodo?
12.2　Ignorancepriors
12.3　Continuousdistributions
12.4　Transformationgroups
12.4.1　Locationandscaleparameters
12.4.2　APoissonrate
12.4.3　Unknownprobabilityforsuccess
12.4.4　Bertrandsproblem
12.5　Comments
13　Decisiontheory，historicalbackground
13.1　Inferencevs.decision
13.2　DanielBernoullissuggestion
13.3　Therationaleofinsurance
13.4　Entropyandutility
13.5　Thehonestweatherman
13.6　ReactionstoDanielBernoulliandLaplace
13.7　Waldsdecisiontheory
13.8　Parameterestimationforminimumloss
13.9　Reformulationoftheproblem
13.10　Effectofvaryinglossfunctions
13.11　Generaldecisiontheory
13.12　Comments
13.12.1　Objectivityofdecisiontheory
13.12.2　Lossfunctionsinhumansociety
13.12.3　AnewlookattheJeffreysprior
13.12.4　Decisiontheoryisnotfundamental
13.12.5　Anotherdimension?
14　Simpleapplicationsofdecisiontheory
14.1　Definitionsandpreliminaries
14.2　Sufficiencyandinformation
14.3　Lossfunctionsandcriteriaofoptimumperformance
14.4　Adiscreteexample
14.5　Howwouldourrobotdoit?
14.6　Historicalremarks
14.6.1　Theclassicalmatchedfilter
14.7　Thewidgetproblem
14.7.1　SolutionforStage2
14.7.2　SolutionforStage3
14.7.3　SolutionforStage4
14.8　Comments
15　Paradoxesofprobabilitytheory
15.1　Howdoparadoxessurviveandgrow?
15.2　Summingaseriestheeasyway
15.3　Nonconglomerability
15.4　Thetumblingtetrahedra
15.5　Solutionforafinitenumberoftosses
15.6　Finitevs.countableadditivity
15.7　TheBorel-Kolmogorovparadox
15.8　Themarginalizationparadox
15.8.1　Ontogreaterdisasters
15.9　Discussion
15.9.1　TheDSZExample#5
15.9.2　Summary
15.10　Ausefulresultafterall?
15.11　Howtomass-produceparadoxes
15.12　Comments
16　Orthodoxmethods：historicalbackground
16.1　Theearlyproblems
16.2　Sociologyoforthodoxstatistics
16.3　RonaldFisher，HaroldJeffreys，andJerzyNeyman
16.4　Pre-dataandpost-dataconsiderations
16.5　Thesamplingdistributionforanestimator
16.6　Pro-causalandanti-causalbias
16.7　Whatisreal，theprobabilityorthephenomenon?
16.8　Comments
16.8.1　Communicationdifficulties
17　Principlesandpathologyoforthodoxstatistics
17.1　Informationloss
17.2　Unbiasedestimators
17.3　Pathologyofanunbiasedestimate
17.4　Thefundamentalinequalityofthesamplingvariance
17.5　Periodicity：theweatherinCentralPark
17.5.1　Thefollyofpre-filteringdata
17.6.　ABayesiananalysis
17.7　Thefollyofrandomization
17.8　Fisher：commonsenseatRothamsted
17.8.1　TheBayesiansafetydevice
17.9　Missingdata
17.10　Trendandseasonalityintimeseries
17.10.1　Orthodoxmethods
17.10.2　TheBayesianmethod
17.10.3　ComparisonofBayesianandorthodoxestimates
17.10.4　Animprovedorthodoxestimate
17.10.5　Theorthodoxcriterionofperformance
17.11　Thegeneralcase
17.12　Comments
18　TheApdistributionandruleofsuccession
18.1　Memorystorageforoldrobots
18.2　Relevance
18.3　Asurprisingconsequence
18.4　Outerandinnerrobots
18.5　Anapplication
18.6　Laplacesruleofsuccession
18.7　Jeffreysobjection
18.8　Bassorcarp?
18.9　Sowheredoesthisleavetherule?
18.10　Generalization
18.11　Confirmationandweightofevidence
18.11.1　Isindifferencebasedonknowledgeorignorance?
18.12　Camapsinductivemethods
18.13　Probabilityandfrequencyinexchangeablesequences
18.14　Predictionoffrequencies
18.15　One-dimensionalneutronmultiplication
18.15.1　Thefrequentistsolution
18.15.2　TheLaplacesolution
18.16　ThedeFinettitheorem
18.17　Comments
19　Physicalmeasurements
19.1　Reductionofequationsofcondition
19.2　Reformulationasadecisionproblem
19.2.1　SermononGaussianerrordistributions
19.3　Theunderdeterminedcase：Kissingular
19.4　Theoverdeterminedcase：Kcanbemadenonsingular
19.5　Numericalevaluationoftheresult
19.6　Accuracyoftheestimates
19.7　Comments
19.7.1　Aparadox
20　Modelcomparison
20.1　Formulationoftheproblem
20.2　Thefairjudgeandthecruelrealist
20.2.1　Parametersknowninadvance
20.2.2　Parametersunknown
20.3　Butwhereistheideaofsimplicity?
20.4　Anexample：linearresponsemodels
20.4.1　Digression：theoldsermonstillanothertime
20.5　Comments
20.5.1　Finalcauses
21　Outliersandrobustness
21.1　Theexperimentersdilemma
21.2　Robustness
21.3　Thetwo-modelmodel
21.4　Exchangeableselection
21.5　ThegeneralBayesiansolution
21.6　Pureoutliers
21.7　Onerecedingdatum
22　Introductiontocommunicationtheory
22.1　Originsofthetheory
22.2　Thenoiselesschannel
22.3　Theinformationsource
22.4　DoestheEnglishlanguagehavestatisticalproperties?
22.5　Optimumencoding：letterfrequenciesknown
22.6　Betterencodingfromknowledgeofdigramfrequencies
22.7　Relationtoastochasticmodel
22.8　Thenoisychannel
AppendixA　Otherapproachestoprobabilitytheory
A.1　TheKolmogorovsystemofprobability
A.2　ThedeFinettisystemofprobability
A.3　Comparativeprobability
A.4　Holdoutsagainstuniversalcomparability
A.5　Speculationsaboutlatticetheories
AppendixB　Mathematicalformalitiesandstyle
B.1　Notationandlogicalhierarchy
B.2　Ourcautiousapproachpolicy
B.3　WillyFelleronmeasuretheory
B.4　Kroneckervs.Weierstrasz
B.5　Whatisalegitimatemathematicalfunction?
B.5.1　Delta-functions
B.5.2　Nondifferentiablefunctions
B.5.3　Bogusnondifferentiablefunctions
B.6　Countinginfinitesets?
B.7　TheHausdorffsphereparadoxandmathematicaldiseases
B.8　WhatamIsupposedtopublish?
B.9　Mathematicalcourtesy
AppendixC　Convolutionsandcumulants
C.1　Relationofcumulantsandmoments
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