信息理论和 知识集聚.docx
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信息理论和 知识集聚.docx
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信息理论和知识集聚
InformationTheoryandKnowledge-Gathering
信息理论和知识集聚
ABSTRACT
Itisassumedthathumanknowledge-buildingdependsonadiscretesequentialdecision-makingprocesssubjectedtoastochasticinformationtransmittingenvironment.ThisenvironmentrandomlytransmitsShannontypeinformation-packetstothedecision-maker,whoexamineseachofthemforrelevancyandthendetermineshisoptimalchoices.Usingthissetofrelevantinformation-packets,thedecision-makeradapts,overtime,tothestochasticnatureofhisenvironment,andoptimizesthesubjectiveexpectedrateof-growthofknowledge.Thedecision-maker’soptimalactions,leadtoadecisionfunctionthatinvolveshisviewofthesubjectiveentropyoftheenvironmentalprocessandotherimportantparametersateachstageoftheprocess.Usingthismodelofhumanbehavior,onecouldcreatepsychometricexperimentsusingcomputersimulationandrealdecision-makers,toplayprogrammedgamestomeasuretheresultinghumanperformance.
KEYWORDS
decision-making,dynamicprogramming,entropy,epistemology,informationtheory,knowledge,sequentialprocesses,subjectiveprobability
摘要
人们一般认为人类知识构造是基于一些互相分离的不相关的决策程序,
ThinkingAboutKnowledge
Chebyshev,astranslatedbyKhinchin(1957),expressedthemagicalrelationshipbetweenscientifictheoryandpractice:
“Thebringingtogetheroftheoryandpracticeleadstothemostfavorableresults;notonlydoespracticebenefit,butthesciencesthemselvesdevelopundertheinfluenceofpractice,whichrevealsnewsubjectsforinvestigationandnewaspectsoffamiliarsubjects.”Anobjectiveforascientististounderstandthephenomenaofhisorherenvironment.Oneofthemeansforunderstandingistobuildhypotheticalathematicalmodelsthatreflectourviewofrealityandhopefullyreachthelawsofnature.Withoutthisunderstanding,howarewetoconductexperimentstovalidateourhypothesisandthenreachoutforevenmoretruthsaboutnature?
Theriskforascientificmodelbuilderistobeabletosteeranarrowpathbetweenthenaivetyofoversimplificationandthemorassofovercomplication.Thesubjectofknowledge-gatheringinhumandecisionmakersisthebasisofthispaper.Recently,thestudyofknowledge-gatheringinmachineshasbeenamajorpartofthenewscienceofArtificialIntelligence.MostofthisAIresearchhasbeentothestudyknowledgeasexpressedalongsemanticorlogicnetworklinestoenableexpertsystems.Knowledgesystems,aslearningsystems,havealsohadalongandsuccessfulhistoryinthefieldofpsychometrics.Theinfluenceofinformationtheoryonknowledgegatheringsystemsthateffecthumandecision-makinghasresultedintheapplicationofseveralmathematicaldiscretesequentialsystemanalysistechniquesthatstemfromtheclassicalworksofH.Poincaré,G.D.BirkhoffandontotherecentworksofthelateRichardBellman(1960)oftheRANDCorporation.These“dynamicprogramming”techniqueshavealsobeensuccessfullyinappliedtomanyotheriversescientificendeavors.Applyingthesesametechniquestotheanalysisofknowledgegatheringinhumandecision-makersmayappeartobea“tailwaggingthedog”tosome,butsometimesnewideascomebystrangepaths.ThesubjectofwhathumanknowledgeishasalwaysoccupiedtheoldphilosophersfromPlatotoHusserl.Recently,ithasbecomepossibletoanalyzeknowledge-gatheringasadiscrete,sequentialstochastic,adaptivedecision-makingprocess,subjectedtothelawsofprobability,andthestep-bystepactionsofhumandecision-makers,whoinKennethArrow’swords,arelearningbydoing(Arrow(1961).
Inthispaper,initssimplestform,Iwilldevelopthemathematicalstructureofahumanknowledge-gatheringmathematicalprocessofthistype.Despitethesimplicityofthismodel,themathematicaldetailsarestillextensiveand,forthesakeofthebalancebetweenthesimpleandthecomplex,IhavedelegatedthosedetailstotheAppendix,hereinafter.
Observationofahuman,decision-maker,behavinginaknowledge-gatheringprocess,indicatesthat,ontheaverage,hisintelligenceincreasesovertimeandtendstosimplifythedecision-maker’sgatheringprocessbydecreasinghisrelativeentropy.Thisprocesscanoccurinthefaceofastochasticenvironment,a“deusexmachina”thatfeedsthedecision-makeravarietyofrandominformation-packets.Someoftheseinformationpacketsarevaluabletothedecision-maker,butmostareirrelevant.Somehow,thedecision-makerisabletosort,theserandominformationpacketsintoalistofincreasingacceptancefromtheirrelevanttotherelevant,dependingonthedecision-maker’spreviousexperience,storedinhismemory.Inthisprocess,thedecision-makertendstodecrease,bothhisrelativeentropy(hisuncertaintyorchaos),inthefaceofaworldwherethenetentropyisbelievedtobealwaysincreasing.Non-intelligentdynamicsystems,forexample,Brownianmotionofmicro-particlesinsuspension,donotseemtobehaveinthismanner.Itappearsthatthesenon-intelligentsystemslackameansoforganizingagainst,oradaptingtothehiddenmechanismsintheirenvironment.Forthemostpart,exceptforsomechemicalprocessesthattendoscillatebackandforthfromchaostoorder,withobviousconsequencestotheeffectontheirentropy,thesenon-intelligentprocesses,intheend,alwaysincreasetheirlocalentropy.IlyaPrigogine(1996)hasgivenmanyexamplesofthiskindofpseudolifelikebehavior.
Inthispaper,Ipostulatethatlifelikesystemshaveseveralnecessaryuniquefacilitiestoeffectrelativeentropydecreasesthroughtheirorganizedknowledgegatheringactivity.Thefirstoftheseisthebasicabilitytomakedecisions.Thisability,inthefaceofcomplexworld,enableslivingbeingstoadaptandcontroltheirecisionmakingthroughtheacceptanceorrejectionofrandominformation-packetstransmittedbytheirenvironment.Withinthesesystems,wefindthehumanknowledge-gatheringactivityenablestheoptimalacceptanceorrejectionofrandominformation-packets,receivedintotheirsensesfromtheirimmediatestochasticenvironmentandstoredintheirmind.AsShakespeareaptlyputitforpoets(andmathematicians),
“Andasimaginationbodiesforth
Theformsofthingsunknown,thepoet’spen
Turnsthemtoshapes,andgivestoairynothing
Alocalhabitationandaname.”
“AMidsummerNight’sDream,”
ActV,SceneI.
Ingeneral,humanknowledge-gatheringsystemspossessauniqueandessentialmemorycapabilitythatenablesthemtooptimizetheirpersonalandsocialdecisionmakingbehavior.TheresearchofthelateHansReichenbach(1999)inhisbook“TheDirectionofTime”hasshownthatmemory(ofsomekind)isanessentialelement,forcingtimetoalwaysseemtoflowinanincreasingdirection.Thishumanmemoryisthesecondnecessaryelementthatenablesthehumandecision-maker,toreducehis
relativeentropyandoptimizehisdecisions.Themathematicalmodel,describedherein,isaembodimentofhowthesekeydependences,decision-makingandmemory,enablethehumanknowledge-gatheringprocessandproduceitsdecreasingeffectonhisrelativeentropy.
Ittakesnerveforaneconomistandcomputerscientisttoleapintoaworlddominatedbycognitivescientistsandphilosophers,andworseyet,proposeyetanothermathematicalmodelforman’scapabilitytoaccumulateknowledge.Mypreviousresearchinmathematicalstatistics,informationtheoryandeconomicdecisionprocessesforcedmepassdownthisdangerouspath.ItallstartedwhenClaudeShannon(1948)developedanoriginalidea.Inhispaper,heintroducedarationalmeasureforinformationbasedontheneedsofthecommunicationengineersatBellTelephoneLabsandthuslaunchedtheformalideaswenowfondlycall“InformationTheory.”Inhisoriginalconcept,Shannonconsideredallinformation-packetshadequalvalue.Sincehisprimaryfocuswasontheefficiencyofthetransmissionofelectronicinformationoverabandwidthlimitedcommunicationschannel,hewasnotdirectlyconcernedwiththevalueofthatinformation.InShannon’soriginalviewthevalueofatransmittedinformationpacketwasalwaysintheeyesofthesenderorreceiver-notthecommunicationsserviceprovider.LeonBrillouin(1963)pointedoutthatthevalueofShannon’smeasureofinformationisdependentonthepotentialuseofthatinformationbyadecision-maker.Shannonandhisassociatessoonrectifiedthatomissionintheirlaterpapers.J.Marshack(1960)advancedtheclassicaltheoryofeconomicsbyhisconsiderationtheroleofinformationasanewkindof“factor”intheproductionfunctionandwassoonfollowedbyseveralothersimilarpapers.Recently,KennethArrow(1996)extendedShannon’scostofinformationtheorytoaremarkablelevel.Arrowhasshownhowinformationtheory,appliedtomoderneconomictheory,causesthegenerationofstrongincreasingeconomicreturnstoscaleinproductionthatdominatestoday’sneweconomicgrowthinthis“information”age.VittorioSomenzi(1965)hasextendedtheaspectsofentropyfoundintheinformationtheorytothe“Mind-Body”philosophiccontroversy.Finally,speciallyforthenon-mathematicalphilosopher,FredDretske(1999)hasrecentlydrawntogether,ingreatdetail,alloftheseknowledgeandinformationfragments.Theseneweconomicandphilosophicapplicationsofinformationtheoryprovidethejumpingoffpointforthispaper.
Intheearly1900’sHenriPoincarégavealecturebeforetheSocietedePsychologieinParisonDiscoveryinMathematics.ThislectureformschapterIIIofhisbook,“ScienceandMethod.”Poincaré(1953),agreeingwithHelmholtz,feltthatthephenomenonofknowledgegatheringwascomposedofthreedistinctphases.AfterattendingPoincaré’sfamouslecture,JacquesHadamand(1954)pulled
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