ExplainingBirds’ColorfulPlumageandBeautyPreferencesbyDemandRelationshipswithInformationandConfirmation

MeasureAnalyses

ChenguangLuEmail:Survival99@gmail.com

Abstract:Whydomanymalebirdsdisplayspecificcolorfulpatternsontheirplumage?Thedemand‐relationshiptheoryexplainsthatbeautypreferencesreflecthumanandbirds’desireforapproachingsomeobjects;thesepatternslookbeautifulbecausetheyresembletheiridealfoodsourcesorenvironments.Mutantsthathaveenhancedhumanandbirds’abilityandmotivationinfindingthesewerehenceselectedbynature.Suchbeautypreferencesofthefemalebirdsthenselectedthemaleplumage.Someconvincingphotosareprovided.Astatisticalanalysisaboutinference“Ifabirdhasyellowfeatherthenitprobablylikeseatingnectar”showsthatthisinferencehashigherconfidencelevelorconfirmationmeasureandconveysmoresemanticinformationthaninference“Ifapersonhashightriglyceridethenheprobablyalsohasfattyliver”.

Keywords:Beautypreferences;Sexualselection;Demand‐relationshiptheory;Foodresources;Colorfulplumage;Hypothesistesting;Shannon’sinformation;Semanticinformation,Confirmationmeasure.

1. Introduction

Wefindcertainlythingsbeautifulandsomeugly.Theremaybesocial,educational,andculturalfactorsaffectingsuchpreferencesandtheremayalsoexistlargeinterpersonaldifferencesinsuchpreferences.However,somecommonfactorsatthenaturelevelcannotbedenied.Forexample,mostmenfindthenakedbodiesofwomenintheirfertileagesattractive.Thisbeautypreferencemakesthemseekfertilewomenandmatewiththem.Thishelpsthemtopassontheirgenes.Thenaturalselectionofsuchfitness‐enhancingpreferencesisobvious.However,therearesomeaspectsthatarenotobviousorevenhavebeencontroversialsincethetimeofDarwin,asdiscussedbelow.

Darwinexplains[1]thatfragranceandcolorfulnessofflowersaretoattractinsectstopollinateforthem;goodflavorandcolorfulnessoffruitsaretoattractbirdsandmammalstoswallowtospreadtheirseeds.However,Darwinwaspuzzledbywhyinsects,birdsandmammalshavesuchtastes.Hewrote([1],Ch.6,Sec.8):

“Howthesenseofbeautyinitssimplestform‐‐thatis,thereceptionofapeculiarkindofpleasurefromcertaincolours,formsandsounds‐‐wasfirstdevelopedinthemindofmanandoftheloweranimals,isaveryobscuresubject.Thesamesortofdifficultyispresentedifweenquirehowitisthatcertainflavoursandodoursgivepleasure,andothersdispleasure.Habitinallthesecasesappearstohavecometoacertainextentintoplay;buttheremustbesomefundamentalcauseintheconstitutionofthenervoussystemineachspecies.”

Darwinadvancedsexualselectiontoexplainthecolorfulappearancesofmalebirdsintermsoffemalebeautypreferencesoraesthetictastes [2].Wallaceobjectedstrongly[3,4],sayingthatthisisaddinganadditionalprincipleofbeautyselectionontopofnaturalselection,withoutexplaininghowfemale taste for beauty comes from. Wallace used camouflage and signaling to explain colorfulplumage in birds, used good sense or sign to explain female sexual selection (selecting goodcharacteristics inmales),whiledenyingtheselectionofmaleappearancebythetasteforbeautyofthefemalebirds[5].Later,Fisheradvancedatheoryinsexualselection[6]calledarm‐racetheorytoreconcilethecontradictionbetweenDarwinandWallace[5].ItisFisherwhoinitiatedthelikelihoodmethodforhypothesistesting.

Wallace explains the plumage of female birds anddancing inmanybird species better,whileFisherexplainsthebigtailsinsomebirdspeciesbetter.Therearealsoothertheories[5,7,8]withsomeadvantages.However,morecases(e.g.theocelliintheplumageofPeacocks,theyellowtailofMandarinDucks, the special patterns in the faces of Baikal Teals,Macaws, andKingVultures) aredifficulttobeexplainedbycurrenttheories.

Accordingtobiologicalutilitarianism,thevarioustastes(includingbeautypreferences)ofanimals(includingman)shouldbebeneficialtobiologicalfitness(survivalandreproduction)forthemtosurviveaftertheiremergence.Accordingly,weproposedademand‐relationshiptheoryfortheevolutionoftastes[9‐11].Accordingtothistheory,pleasantsensationsreflectdemandrelationshipsandpromotetheirdevelopments.Senseofbeauty,asoneofpleasantsensations,reflectsandstirsourdesireforapproachingsomeobjectsorenvironmentsthatmaybedirectlyorindirectlybeneficialtooursurvival.Thisevolutionmaybethroughbothnature(biologicalvariation;mutationinparticular)andthroughnurture(environment,education,learning,etc.).

Theco‐evolutionofflowersandinsectsandthatoffruitsandbirdsandanimalsaretheresultsofnature(throughvariationandselection).Wemayimagine[9]thatoriginallyflowerswerenotcolorful,andinsectsdidnothavegoodtastesforflowers.Sincecross‐pollinationisparticularlybeneficial,thoseplantsthatmutatedtohavepollenandnectarneededasfoodbyinsectsgainedanadvantageandprosperedmore.Simultaneously,thoseinsectsthatdevelopedataste(insight,taste,orsmell)forpollenandnectarhadhighermotivationforseekingflowers.Thisledtomutualreinforcementandahighdegreeofco‐evolution,resultinginveryfragrantandcolorfulflowers.Similarly,theemergenceoffragrantandcolorfulfruitswhenripemadethemmoreeasilyfoundandeatenbybirdsandanimals,helpingthemtospreadtheirseeds.Birdsandanimalswithsuchtasteshavehighermotivationandabilityinlocatingthesefruits,leadingalsotoco‐evolution.

Weallfindcowdungugly.However,peopleinTibetuseittodecoratehome1.Thismaybeexplainedbythefactthatthesepeoplehavehadalonghistoryofseekingaftercowdung,usingitasfuel.Thishighdemandmighthavefosteredtheirtasteforbeautyincowdung.

Theevolutionofdemandobserves the ruleofmeans turning into ends [10,11].Forexample,fishingwasthemeans foreatingfish,but ithasbecomeanendin itself;kissingwasthemeansforhaving sex, but has become an end in itself; if the fruits are nice food, their flowers also appearbeautiful.Thisissobecauseitiseasiertoobtainthefruitsinanenvironmentwithsuchflowers.Thisisthereasonformanandanimalsfindingflowersbeautiful.

1 http://lxs.cncn.com/53654/n357640

Theauthorisnotalone.Afterheproposedtheabovetheoryandfoundthatpeacockmimickedtheberrytree[9],MerleE.Jacobsalsoproposedthisfinding[12].Roddandetalfoundthatguppieshadorangeroundspotsontheirtailsformimickingtheirfavoritefoods[13].ThisfindingobtainedveryhighpraisebyapaperonScience[14].

This studywill providemoredetailed evidence to support thedemand‐relationship theory ofbeautypreferences,includingsomephotosandsomestatisticalresults.

2. MaterialsandMethods:UsingFoodSourcesandEnvironmentstoExplainBirds’ColorfulPlumage

Wewishtoadvanceafood‐sourcedemand‐relationshiptheorytoexplaintheparticularpatterns,notonlyinthepeacockbutalsoinmanyotherspecies.Thebasicideaisthis.MembersofaspeciesAprefertoeatacertainfoodsourceX.ThisrelationshipallowsAtoevolvesothatA’smembers(femalemembersinparticular)findingXbeautiful.Oncethisbeautypreferenceisestablished,ifsomemalemembers ofA happen topossess someaspectofX that attracts femalemembers ofA, perhapsbymutation,theywillthenattractmorefemalesofAtomatewiththem.ThistraitofmalemembersofAthatmimicX thengot selected.That thisprocessmaywork is clear.That it in factdidso inmanyspeciesarepresentedbelowontheveryclosesimilarityofthecolorfulpatternsofsomepartsofthemalemembersbeingverysimilartothemainfoodsourcesoftherelevantspecies.

WhydoestheplumageofPeacockscontainthedistinctiveocelli intheirtails?Peafowlsliketoeatberries.This first led to themto findberriesbeautiful.Thisbeautypreferenceofpeahens thenselectedfortheocelliinthetailsofpeacockssothatapeacockdisplaysthetailthatlookslikeaberrytree (seeFigure1). Peacocks display and shake their tailswhile courting, probablymimicking themovementofaberriedtreeinwind.Thereisaparalleltothis;ThePeacockPheasantalsohassimilarplumagewithocellilookingmorelikeballs([2],Chapter13).BothPeafowlsandPeacockPheasantsliketoeatberries.

(a)(b)

Figure1APeacock(a)(ThankstoJeanRogers2)andawildblueberrytree(b)(ThankstoShargaljut3).

2 This image is licence free from https://ccsearch.creativecommons.org/image/detail/uA0Qy6etODx6WWp99i5jFQ== 3 This is a royalty free image from https://www.dreamstime.com/royalty-free-stock-photos-blueberry-bush-image11589318

Wecanfind, inYunnanprovince,China,thattheareawherethepeacockusedtolive[15]andtheareawherethewildblueberrygrows[16]arehighlyoverlapped.Theseareasalsoaboundwithplums. It is verypossible that thePeacockmimics the blueberry tree or theplum tree.We testedpeafowls and pheasants with blueberries and corns. It was obvious the blue berries were moreattractivetopeafowlswhereascornsweremoreattractivetopheasants4.

Similarly,manybirdssuchasBeautifulFruitDovehaveredspotsontopoftheirheadsthatclearlymimicripefruitsthatthosebirdsliketoeat(Figure2).

Figure2.BeautifulFruitDoveswithheadsthatmimictheredfruits.ThankCoolPictureGallery5foritskindpermission

TheCarduelisCarduelishassimilarheadbecausethisbirdalsolikeeatingredfruits.Similarly,aparrotmayhave amango‐likehead6.We canalso find somebirds, eachofwhich seems tohaveafloweronhead,suchasKingletandGoldcrest.Thesebirdsmustlikeflowersforeatingnectar.

Also,manybirdsdisplaypatternsofgrainears intheirplumagebecausethey liketoeatthesegrains(Figure3).Birdswithgrain‐likeplumageincludeMontezumaQuail7，ZebraFinch8,andHouseCock9.

4 More details about peacocks, see http://blog.sciencenet.cn/blog-2056-872375.html 5 This image has licence granted from http://coolhqpix.blogspot.ca/2011/02/worlds-most-beautiful-birds-photos.html 6 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=422991 7 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=427860#pic_block 8 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=440247 9 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=428603#pic_block

(a) (b)

Figure3.AChrysolophuswithgrainearspatternonitstail(a)10andgrainears(b)11

There are alsomany bird species with plumage resembling insects that they like to eat. Forexample,theSunbitternhasplumagemimickingbutterfliesandmoths(Figure4).OnecanfindmanyphotosonlineshowingtheSunbitterneatingbutterfliesandavideoshowingtheprocessofcapturing.The colorful patterns of butterfliesmay be for camouflage or for displaying beauty as thatmimicflowers, henceattracting theopposite sex. Somebutterflieshave ‘eyes’ on theirwings, serving thepurpose of warning against their natural enemies. Besides displaying beauty, the Sunbittern alsodisplaysunfoldedwingstoscareenemieswhenthreatened.

Figure4.ASunbitternthatmimicsabutterfly.ThankstoTomasGrim12forhiskindpermission

Duetothecostsinvolved,themimickingisusuallyonlypartial.Forexample,theBeeEaterdoesnotmimicthewholebeebutjusttheyellowfinehairofabee13;whenthemaleTemminck'sTragopan

10 From licence free website http://www.nipic.com/show/2930954.html. Thank the author. 11 From http://www.sxagri.ac.cn/166/kfbpubzljs.asp?id=504. Thank the website and the author for kind permission. 12 http://ibc.lynxeds.com/photo/sunbittern-eurypyga-helias/one-bird-perching-rock-middle-cristalino-river

mimicsthegrasshopper(orasimilarinsect),onlyitsheadresemblesthegrasshopper’shead14.Thesimulationisrelatedtodisplayingcostandmutationhistory.

Wecanalsofindsomebirdsmimickingantcoloniesoranteggs,suchasBrownThrasher15,WoodThrush, Ovenbird, and Rusty‐backed Antwren. They either like eating ants (or ant eggs), or likefollowingarmyantstofindinsectsforfood.

Mostspeciesofhummingbirdsdisplayflower‐likecolorsandpatternsontheirnecks16becausehummingbirdslikesflowers.Darwinalsodescribedatypeofhummingbird,theLophornisOrnatus,withaflower‐likepatternonitshead([1],Ch.13).TheRainbowLorikeet17feedsmainlyonfruit,pollenandnectar,italsohasmulticoloredplumage.WediscoveredthattheGreaterSpottedWoodpeckerhastheneckwiththepatternlookinglikeuncoveredtunnel;theJynxTorquillahasshapeslookinglikeanttunnels.Allthesemaybeexplainedbythedemand‐relationshiptheory.

TheGreaterBirdofParadisedoesnotonlymimicfliesbutalsomimicflowers18.Thisbirdmustlikeeatingfliesandlivingwithflowers.TheWilsonBirdhasbeetle‐likeandinsect‐likepatternsonheadandback19.Thisbirdmustlikeeatingbeetlesandinsects.Variousbirdsofparadisedisplayveryspecialappearancesandgestures.Wehavetoexaminespecialenvironmentstofindspecialcauses.

Manybirdslivinginwatermimicwater‐relatedfoodsources.Forexample,maleMandarinDuck(YuanYang inChinese)haswings andyellowplumages thatmimic a clamwith stretching flesh20;maleWoodDuck(alsoknownasNorthAmericaYuanYang)haswingslookinglikeshellfishandheadlookinglikesasnail21.MaleKingEideralsohasaheadlookinglikeasnail22.WecanfindthattheKingEiderreallylikeseatingshellfish.Snailsmustbeitsidealfoods.Longtimeago,snailsmustbegoodfoodresourcesfordifferentkindsofducks.

TheTuftedPuffinhasahead looking likeaprawn(Figure5).Weknowthat theTuftedPuffineatsshrimpsorprawnsandcangounderwatermorethan60meters; thespotprawn lives in theareawheretheTuftedPuffinlives.WeguessthattheTuftedPuffinmustlikeeatingthespotprawn.Note that theupperbeak is flat,yet it looksround likeaprawn’sbackverymuch.The lowerbeaklookslikeaprawn’stail.Theseareunlikelytobecoincidental.Thewhitefeatherisnotidealtomimicaprawn’shead.Perhapsitmimicsasquid’shead.There isaclearerphoto23.WatchingtheAtlanticPuffin from the front,we can find this bird has a Surf Clam‐like beak24. Surf clams happen to livewhere Atlantic puffins often appears and spot prawns happen to live where tufted puffins oftenappear.Aretheseoccasional?

13 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=410908 14 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=8516 15 https://en.wikipedia.org/wiki/Brown_thrasher 16 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=422990 17 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=601203 18 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=601208 19 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=326538 20 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=8515 21 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=8513 22 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=blog&id=742139 23 http://stock.chrisdoddsphoto.com/famain.asp?customerId=112&sKey=GHCSEHNJ&action=viewimage&cid=76&lid=0&imageid=283 24 http://img3.imgtn.bdimg.com/it/u=1808009408,2066031166&fm=214&gp=0.jpg

Fig 5. A Tufted Puffin with beak that mimics a prawn’s body and tail. Thanks to the creator Alan D.Wilson25.

Two good examples supportive of the demand‐relationship theory of taste origin are theMacawand theKingVulture. Theyhavebeautypreferences quitedifferent fromours. Though theplumageofMacawsisbeautifultous,theclay‐likepatternsontheirfaceslookugly(seeFigure6;seethispage26formoredetailedcheek).Why?Todispensewiththepoisoncontainedintheleavesthatthey eat, Macaws also eat clay. This demand for clay makes them find clay beautiful and theappearanceofclay‐likespots(inbothcolorandpattern)ontheirfacegotselected.

25 This image with freen licence is from https://en.wikipedia.org/wiki/Tufted_puffin 26 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&picid=410907

Figure6.Macawswithcheeksthatmimicclay.ThankstoBrainRalphs27

Similarly,astheKingVulturelikestofeedondeadbodiesoflargeanimals,maleKingVultures

displaystornfuranduncoveredflesh(Figure7).TheRedHeadedVulture28whichalsofeedondeadbodiesdisplaysthesefeaturesnotonlyonitsfacebutalsoonitsbody.

27 ByBrianRalphs‐originallypostedtoFlickrasDSC_8774.NEF,CCBY2.0,https://commons.wikimedia.org/w/index.php?curid=9678869.28 http://www.indianaturewatch.net/displayimage.php?id=354041

Figure7. AmaleKingVulturedisplaysthepatternwithtornfuranduncoveredfleshonneckandgutonnose.ThankstoOlafOlivieroRiemer29

Thefood‐sourcetheorymaybeexpandedtoincludethesustaining‐environmenttheory.Here,anenvironmentXsustainsthelifeofspeciesAwell.ThisrelationshipallowstheevolutionofAtohaveabeautypreferenceforXasthisallowsmembersofAtofindXbetter.SomepatterninthebodyofthemalemembersthatemergefrommutationthatlookssimilartoXthengotpreferentiallyselectedbythepreferenceoffemalemembers.

Manyspeciesofduckshave,ontheirplumage,blueorgreenpatternmimickingdeepwater,wavypatternmimickingshallowwater,andbrownpatternmimickingwetland.Forexample,theGreenWingedTealhaveeachofthesepatterns(Figure8).

Figure8.Amalegreenwingedteal.ThankstoCharlesandClint30

TheBaikalTealhasmoredelicateisland‐likepattern(Figure9).Inthewinter,BaikalTealsflytotheYangtzeriverbasin.Sandislandsoralluvionsintheriveraretheirmostfavoredspotsforfeedingandresting(beingawayfrompredators).Themimickingofalluvionsontheirfacesisvery

29 This image with CC licence is from https://en.wikipedia.org/wiki/King_vulture.

30 This image with CC licence is from http://www.flickr.com/

impressive.Notetherearegreencolorforriver,yellowcolorforalluvions,blackcolorforriverbank,andwhitecolorforspoondrift.

Figure 9. A male Baikal Teal with the head that displays alluvions on a river. Thanks to Lool for his kindpermission31.

Similarly,theEgyptianPlover,whichbreedsonsandbarsorallucionsinverylargerivers,alsodisplaysriverandalluvion32.Asislandsprovidesafehavenswithfoodsforbirds,thosebirdsfindingisland‐likepatternsbeautifulprefertochooseislandstolandontheirflightsandsurvivebetter.Thus,manybirds(e.g.Red‐breastedGoose33，SpectacledEider34，Bufflehead35)haveisland‐likepatterns.Theisland‐likepatternsinarcticbirdsareusuallywhite,whichexactlymimicsnow.

Therearesomebirds(e.g.mostgrousesandsomebirdsofparadise)whoseappearancesmaybedifficulttoexplainintermsoftheirfoodsourcesorlivingenvironments.Thereareatleasttworeasonsforthis.First,wemaynotknowtheirhabitsenough.BeforeweknowthatMacawseatclay,itwouldbedifficulttoexplaintheuglyclay‐likepatternandcolorontheirfaces.Secondly,thepreferredfoodsourcesofsomespeciesmayhavedisappearedsometimeago.

3. InformationandConfidenceLevelAnalyses

3.1 FromShannon’sInformationTheorytoSemanticInformationTheory

We need to measure the semantic information and the confirmation measure of a hypothesis. For this purpose, we simply introduce how Shannon’s information theory is generalized into semantic information theory. Shannon’s mutual information [17] is defined with

( | )( ; ) ( , ) log

( )i j

i jj i i

P x yI X Y P x y

P x (1)

Where X is a random variable representing the source, and x1, x2, … are its possible values; Y a random variable representing the destination, and y1, y2, … are its possible values. If Y=yj, then we have the Kullback-Leibler information [18]:

31 http://www.cnbird.org.cn/ (search “BaikalTeal”canfindthecreatorLoolandhisworks)32 http://en.wikipedia.org/wiki/Egyptian_Plover 33 http://tolweb.org/Branta_ruficollis/89154 34 https://en.wikipedia.org/wiki/Spectacled_eider 35 https://en.wikipedia.org/wiki/Bufflehead

( | )

( ; ) ( | ) log( )

i jj i j

i i

P x yI X y P x y

P x (2)

When Y=yj, the above equation becomes

( | ) 1( | ) 1( ; ) log log

( ) ( )i jP x yi j

i ji i

P x yI x y

P x P x (3)

Extending Shannon’s information theory, we can obtain the semantic information theory [19-21]. We use θ to denote a predictive model (or model parameters), use θj to denote a sub model associated with yj, and use T(θj|X) to denote the truth function of a hypothesis yj. Using T(θj|X) and P(X), we can obtain a

likelihood function [19]：

( | ) ( ) ( | ) / ( ); ( ) = ( ) ( | )i j i j i j j i j ii

P x P x T x T T P x T x (4)

Replacing P(xi|yj) on the right of the log in Eq. (2) with P(xi|θj), we obtain the semantic Kullback-leibler information

( | ) ( | )( ; ) ( | ) log = ( | ) log

( ) ( )i j j i

j i j i ji ii j

P x T xI X P x y P x y

P x T

(5)

which is compatible with Popper’s theory well [21, 22]. To average I(X; θj) for different yj, we have the semantic mutual information

( | )( ; ) ( , ) log

( )j i

i jj i j

T xI X P x y

T

(6)

It is easy to prove [21] that when

P(X|θj)=P(X|θj) or T(Aj|X)∝P(yj|X) (7)

I(X; θj) reaches its upper limit: Kullback-leibler information I(X; yj). If for each yj, P(X|θj)=P(X|θj), then the semantic mutual information I(X; θ) reaches its upper limit: Shannon’s mutual information I(X; Y). Dividing the two sides of Eq. (6) by P(X), we have the optimized truth function

T*(θj|X)/T(θj)=P(yj|X)/P(yj) (8)

Since the maximum of T*(θj|X) should be 1, replacing T*(θj|X) with 1 in the above equation, we obtain T(θj)/P(yj)=1/P(yj|xj*) where xj* is the xi that makes P(yj|xi) be the maximum of P(yj|X). Them Eq. (7) becomes

T*(θj|X) = P(yj|X)/P(yj|xj*) (9) P(yj|X), j=1, 2, … form a Shannon channel and T(θj|X), j=1, 2, … form a semantic channel. The above formula shows that when the semantic channel matches the Shannon channel, the semantic mutual information is equal to the Shannon mutual information.

3.2 TheConfirmationmeasureofStatisticalInferencefromYellowtoNectar

Wecanuseinformationcriterion,whichwasinitiatedbyPopper[22],toevaluateaprediction,suchas“TuftedPuffinscancatchandliketoeatspotprawns”.Thepriorprobabilityinwhichabirdcancatchspotprawnsthatliveunderwatermore60metersisverysmall,say1/500.Iftuftedpuffinsreallycan,theinformationconveyedbythepredictionshouldbelog2500bitsaccordingtoEq.(3).Wecanusethismethodtoestimatetheamountofinformationofvariouspredictionsaboutbirds’foodresourcesandidealenvironmentsaccordingtobirds’appearances.WecanalsousestatisticaldatatoobtaintheaveragesemanticinformationandconfirmationmeasureofapredictionaboutbirdsaccordingSection3.1.

Wefoundthatbirdswithwavypatternweremostlikelytoliketodrinkwaterfrequently(suchasseveralkindsofsandgrouses36)orliketoliveinorbesidesshallowwater(suchasmanykindsofducks);Birdswithyellowfeatherweremostlikelytolikeeatingnectar(includingpollen).Nowweusesemanticinformationmethodtocalculatetheconfirmationmeasureofinference“Ifabirdhasyellowfeatherthenitprobablylikeseatingnectar”tosupportthedemand‐relationshiptheory.Camouflagemaybeconsideredareason;however,mostflowershaveredpetalswithyellowstamens(orpistils).So,camouflageisanotagoodexplanationforyellowfeather.

Figure10.ASunBirdwithyellowfeatheriseatingnectar.ThankstoPsJeremy37

Weusex1todenoteabirdwithyellowfeatherandx0todenoteabirdwithoutyellowfeather,andusey1todenote“Xeatsnectar”(Xtakesvaluex0orx1)andusey0todenote“Xdoesnoteatnectar”.Thetruthfunctionofinference“Ifabirdhasyellowfeatherthenitprobablylikeseatingnectar”isT(θ1|X)whereθ1isapredictivemodel(submodelofθ)associatedwithy1.Theθ1canalsoberegardedasafuzzyset[21]whoseelementsarebirdsthatlikeeatingnectar.

Thereare843representativebirdsinthebook[24].Thebookshowsthecolorpictureandhabitincludingfoodpreferenceofeachbird.WeroughlyclassifiedthemintofourkindsasinTable1.Forsomebirds,wealsomadeuseofsupplementaryinformationfromsomewebsites.

36 https://en.wikipedia.org/wiki/Sandgrouse

37 This image with licence granted is from https://ccsearch.creativecommons.org/image/detail/1xRWOpDNb4In4apQG-HMiA==

Table1.Thenumbersof4kindsofbirdsrelatedtoyellowandnectar

Numbersofbirds Withoutyellow(x0) Withyellow(x1) TotalNoteat(y0) 686 57 743Eatnectar(y1) 17 83 100Total 703 140 843

TheconditionalprobabilitiesandKullback‐LeiblerinformationaccordingEq.(2)areshowninTable2.

Table2.Shannon’schannelP(Y|X)andKullback‐Leiblerinformationbetweenyellowandnectar

x0 x1 P(yj) Information

y0 0.976 0.407 0.88 I(X;y0)=0.050bit

y1 0.024 0.593 0.12 I(X;y1)=1.537bit

ToaverageI(X;yj)fordifferentyj,wecanobtainShannon’smutualinformationI(X;Y)=0.226bit(accordingtoEq.(1)).

Forauniversalhypothesis,itsfourtruthvaluesareT(θ1|x1)=T(θ0|x0)=1,T(θ1|x0)=b1’(disconfirmationmeasureofy1),andT(θ0|x1)=b0’(disconfirmationmeasureofy0).Theoptimizeddisconfirmationmeasures

b1’*=P(y1|x0)/P(y1|x1)andb0’*=P(y0|x1)/P(y0|x0)(10)

andsemanticKullback‐LeiblerinformationI(X;θ1)andI(X;θ0)areshowninTable3.Theb1’*isalsothesignificancelevelofy1.

Table3.SemanticchannelT(θ|X)*andsemanticKullback‐Leiblerinformationaboutbirds

x0 x1 T(θj) I(bit)

y0 1 b0’*=0.417 0.903 I(X;θ0)=0.05y1 b1’*=0.041 1 0.200 I(X;θ1)=1.54

*θisarandomvariableforapredictivemodel.

ThesemanticmutualinformationI(X;θ)isalso0.226bit.Thatmeans,withtheoptimizedconfirmationmeasures,wecanmaximizethesemanticmutualinformationuptoitsupperlimit:Shannon’smutualinformation.Theconfirmationmeasureofy1shouldbeb1*=1‐b1’*=1‐0.04=0.96.

Theoptimizedconfirmationmeasureb1*isequalto1‐1/LR+[19](LR+istheMaximumLikelihoodRatioinmedicaltests[25]).Themaximumsemanticinformationcriterionisequivalenttothemaximumlikelihoodcriterionandthemaximumlikelihoodratiocriterion[21].Inmedicaltests,LR+indicateshowgoodatest‐positiveis.Obviously,b1*hassimilarsignificance;yet,itismoreunderstandable.Theconfirmationmeasureb*isalsosimilartoConfidenceLevel(CL).Thedifferenceisthatb*maybenegative.Thenegativeb*meanstheoppositehypothesisissupportedbytheevidence.WeuseCL’=1‐CLtodenoteno‐confidencelevel,thenthereis[21]

1 '/ , if 0.5

'/ 1, if >0.5

CL CL CLCM

CL CL CL

(11)

Tounderstandtheinformationandconfirmationmeasureoftheaboveinferencebetter,wecomparetheaboveinferencewithinference“Ifapersonhashightriglyceridethenheprobablyalsohasfattyliver”.

Thestatisticaldata[25]fromhealthexaminesof500peopleshowtherelationshipbetweentriglycerideandfattyliver.Similarly,wecancalculatetwooptimizeddisconfirmationmeasuresandtwoamountsofsemanticKullback‐LeiblerinformationasshowninTable4.

Table4.Semanticinformationanddisconfirmationmeasuresofinferencesbetweentriglycerideandfattyliver

nofattyliver(x0) fattyliver(x1) lowtriglyceride(y0) 3223 427 P(y0)=0.73hightriglyceride(y1) 647 703 P(y1)=0.27T(θ0|X) 1 b0’*=0.454 I(X;θ0)=0.057bitT(θ1|X) b1’*=0.269 1 I(X;θ1)=0.121bit

Comparingb1’*inTable3andb1’*inTable4,wecanfindthatinference“Ifabirdhasyellowfeatherthenitprobablylikeseatingnectar”hashigherconfirmationmeasureandconveysmoreinformationthaninference“Ifapersonhashightriglyceridethenheprobablyalsohasfattyliver”.Thehypothesisthatdemand‐relationshipsselectsbeautypreferences,asanuncertainhypothesis,issupportedtoconsiderableextentbytheabovestatisticalanalyses.

4. DiscussionsandConclusions

Wemayaskmoreultimatequestions.Obviously,thecolorfulflowersaretoattractinsectstohelpthemcross‐pollinateandthebeautifulplumageofmalebirdsistoattractfemalebirds.Inneitherofthesecasesaretheymeanttoattracthumans.Amoreultimatequestionisthus:Whydowehumansalsofindmostoftheseflowersandplumagebeautiful?(Anexceptionthatprovestheruleistheuglyfacesofvulturesthatfeedondeadbodies,asdiscussedabove.)Wemayventuretoextendourdemand‐relationshiptheoryfurther.Birdsandhumanshavesomecommonfoodsourceslikeripefruits.Thismakesbothbirdsandhumansfindripefruitsbeautiful.Byextension,thismakesusfindtheplumageofmostbirdsbeautiful.Manyflowersarebeautifultohumansbecausetheybearediblefruits.Humansalsofindmanyflowersnotbearingediblefruitsbeautifulbecausetheyaresimilartootherflowersandshowfertileenvironments.Morediscussionsabouthumanbeautypreferencescanbefoundin[11].Wewishtheaboveexplanationscanattractmoreresearcherswhomayyieldmoreinsights.

Wehaveprovidedultimateexplanationsfortheoriginofparticularbeautypreferences.Inparticular,theoriginofcertainpatterns/colorsinbirdsintermsoftheirfooddemandandlivingenvironmentisexplainedwithsomeconvincingphotos.Ourdemand‐relationshiptheorymaybesubjecttorigoroustest.Wehaveexaminedmanypredictions,accordingtobird’sappearances,suchastheBaikalTealandtheEgyptianplover’slikingalluvionsinrivers,theSunbittern’slikingeatingbutterflies,theKingEider’slikingeatingsnails,theMacaw’slikingeatingclay(Therearemorephotos38).Wealsousestatisticaldatatocalculatetheconfirmationmeasureofhypothesis“Ifabird

38 http://blog.sciencenet.cn/home.php?mod=space&uid=2056&do=album&id=23161

hasyellowfeatherthenitprobablylikeseatingnectar”,whichishighupto0.96andstronglysupportourdemand‐relationshiptheory.Westillneedtofurtherexaminesomepredictions,suchastheTuftedPuffin’slikingeatingprawns,theMandarinDuck’slikingeatingclams.Wealsofindsomebirds,suchastheSageGrouse,withpeculiarappearancesthatarehardtobeexplainedbythedemand‐relationshiptheory.Partlybecauseofexpertiselimitationsandpartlyforthesakeofecologicalchanges,wecanonlyleavethesetootherresearchers.

AcknowledgementTheauthorthanksProfessorYew‐KwangNg(whoisafamouseconomistatNanyangTechnologicalUniversity.Hisstudiesonwelfareeconomicswithhappinesscriterionareveryattractiveandhelpfultotheauthor)forhisencouragementandhelpinwritingthispaper.

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