Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy

Absrmez-Jaynes’s principle of m@mum entropy and KuUba&‘s priodple of min imum cromentropy (mhlmum dire&d dfvergenoe) are shown tobeunfquelycomxtmethodsforhductiveinf~whennewinformnt ionlsghninthefomlofexpe&edvalues.ReviousjILstit icatioaslLve htuiUvearguma@andrelyonthepro~ofentropyanderossenhppy lasinfmersmes.‘Illeapproarbben?that reasoMble methodsofhductiveinfere~shouidleedto~nsistentresuitswhentbere are different ways of taking the SIUM! information into acumnt (for eh different cJxKdwesystems).‘lhlp~isf~ asfourc?xn&tencDxlomaThesearestptedhtenmofon~ infonMt&noperPtMaIldmakenoref~ncetoinf~~Itb ~v~tbrttheprincipleofmaximumentropyheorrectinfbef~~ seme:~maxhizhnanylunctioobutentropywiUleodto~ unless that funetkm and entropy hnve identkal maxima. In other vvurds, givenlmformnthintheformofconstraintsonenrpeetedvahres,tbereis onlyonedWibuthsat&fyingthecxnstnhtstbatcmnberhosenbya pJ-odmtbat-the ~axioms;tbiSunlquedisMbUtlOIl cmnbeobtahedbymaximh@ entropy. This result is established both dlrectly~asespedalcase(unilormpriors)dmanalogousresultfor tbeprindpkofndnimum crossentropy. Results are obtahed both for eoldmow probabuity dell&lea and for d&Crete distributioas.