概率论沉思录

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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