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cmbx10ÆAbstract.Ž©E{‘JKéóKñ`y
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Next,‘Ÿ—wš¸ãe“em˜bGed‘r'this“m˜ultiscale‘r'moGdel“in˜to‘r'a“probabilistic/lik˜eliho•Go“dŽ¡‘4eframewš¸ãork–¸(via“hierarc˜hical‘·Ba˜y˜es),‘Øallo˜wing“us‘·to“handle“statistical“uncertain˜ties.‘Z=W‘ÿ*ªe“thenŽ¡‘4euse–ÖeMCMC‘ÖDto“map“out“the“shapšGe“of“the“join¸ãt‘Öfp˜osterior“probabilitš¸ãy“of“all“of“the“unkno˜wnŽ¡‘4eparameters.‘|#Finally‘ÿ*ª,‘.ûw¸ãe–tnote›sthat“this˜is“a“generalization˜of“a“problem˜with“a˜kno¸ãwn“so-Ž¡‘4elution:‘lÆwhether–ÒÕan“additional“compGonenš¸ãt“of‘ÒÔkno˜wn“shapGe“is“justied‘ÒÔb˜y“the“data.‘êFHence,Ž¡‘4ewhen–njthe‘nimš¸ãultiscale“structure“that“w˜e›niuse“to“moGdel˜the“\mismatcš¸ãh"“bGet˜w˜een“the‘nidata“andŽ¡‘4ethe–·5phš¸ãysical“moGdel“accoun˜ts“for“a“signican˜t“n˜um˜bGer“of“coun˜ts,‘ÖÕand/or“when“the“scale-factorŽ¡‘4efor–òˆthe›ò‰bGest-t“ph¸ãysics˜moGdel“is˜signicanš¸ãtly“dieren˜t“than›ò‰one,‘Kthen“the˜original“t˜w¸ãas“notŽ¡‘4e\go•Go“d-enough".‘i
That–;'is,‘@cthe“tted“mš¸ãultiscale“\mismatc˜h"“isolates“the“discrepšGency“b˜et•¸ãw“eenŽ¡‘4emoGdel›‡äand–‡ãdata.‘-MW‘ÿ*ªe“use˜moGdels˜of˜the“Gamma-Ra¸ãy˜sky˜as“view•¸ãed˜b“y˜CGR“O/EGRET‘‡¯as˜ourŽ¡‘4eexample.‘¥=W‘ÿ*ªe–»Òdemonstrate“that›»Ñour“methoGd“w¸ãorks˜nicely“on“sev¸ãeral“examples,‘Õpbut“furtherŽ¡‘4ewš¸ãork–UUis“needed“to“in˜v˜estigate“the“methošGd's“p˜o•¸ãw“er.Ž¦‘JKéThe–7–authors›7•and“CHASC‘7Žac•¸ãkno“wledge–7–suppGort“from˜NSF‘7Ž(DMS‘704-06085)˜and“Chan-Ž¡‘4edra–-ÿX-Raš¸ãy“Cen˜ter,‘5Ýand“AISRP‘-õNA˜G5-12082.‘d«It“is“a“particular“pleasure“to“ac˜kno˜wledge“theŽ¡‘4eSAMSI06–UUSpGecial“Program“in“Astrostatistics.ŽŸ1¥û‘à
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analysis:‘®è\Is–H{a“tted“moMÞdel“ÄGo–ÿp¹o“d‘”ƒEnough¹?"‘¾and–H{\Is“there“additional‘H|unkno²!wn“structure“inŽ¡‘à
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the–o—probabilitš²!y“space“around“the“maxim˜um“lik˜eliho•MÞo“d‘o˜estimate)–o—and“ÊQuestion“¹2“(suc˜h“asŽ¡‘à
computing–ïXlimits›ïWon“the“size˜of“the“pMÞossible˜additional“compMÞonen²!t,‘”or“calibrating˜the“riseŽ¡‘à
in›G
log-lik²!eliho•MÞo“d–Gas˜in˜Protasso²!v˜et“al.‘¾(2001)˜and˜the“references˜therein).‘¾Ho•²!w“ev“er˜outsideŽ¡‘à
the–P`Gaussian“realm,‘zÞÊQuestions“¹3‘Paand“4|is“the“tted“ph²!ysical“mošMÞdel“go˜o˜d“enough“andŽ¡‘à
hoš²!w–jçdoMÞes‘jèone“quan˜tify“additional“structure›jèin“the“image|are˜tough.‘ÊSince“w²!e“will˜use“theŽ¡‘à
structure–èlater,‘øƒwš²!e“next“sk˜etc˜h“four‘èexample“problems,‘ø‚rather“pMÞedan˜tically‘ÿe,‘øƒfrom“simplestŽ¡‘à
to–¦fmore“troublesome.Ž©[‘à
Å1.1.Ž‘4JOne–2Bin“ProblemsŽŸå¿‘à
ÊPršÁ´oblem–"ù1:‘jOne“Bin,“Ga˜ussian“St–ÿEa“tistics.Ž‘à
Ÿº¥‰ fe æ}XŽ’Pš¹In–Üthe“rst“problem,‘7ùdata“are“a“rate“inŽŸ
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ing–!öcompšMÞonen²!t‘!õcan“b˜e“estimated“simply›!õas“the“data“min²!us˜the“moMÞdel“(i.e.,‘@Ùthe“residual),Ž¡‘à
whether–TAone›TBprefers“a˜mean,›d¯a“moMÞde,˜or“a‘TBmedian,˜since“they‘TBare“the“same.‘Â|Unfortunately‘ÿe,Ž¡‘à
the–¦fsimplicit²!y“of“this“analysis“is“unique“to“this“simple“case.ŽŸ£‘à
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Ÿº¥‰ fe ÑMûŽ’ ð!=¹In–‘*this“case,‘ËÛthe“data“are“again“in“a“singleŽ¡‘à
bin,‘0Pwith–Êthe“knoš²!wn“ph˜ysical‘ËmoMÞdel“giving“the“mean.‘¬ªBut“no˜w“the“\rate"“is‘Ëa“discrete“v‘ÿdDari-Ž¡‘à
able,‘áŽi.e.,‘áthe–Õ¹Pš²!oisson-distributed“coun˜ts“in“the“bin“(or“detector)“of“kno˜wn“size“(or“kno˜wnŽ¡‘à
\eectivš²!e–
2area").‘AOnce“again,‘&äthe“computation“of“a“tail“probabilit˜y“is“straigh˜tforw˜ard“(ifŽ¡‘à
more–a÷tedious|it“m²!ust‘aöbšMÞe“calculated“rather“than“lo˜ok²!ed“up;‘¿¿see‘aöBevington“1969,‘ÛLoredoŽ¡‘à
1992).‘XáOne–zhas“a“cš²!hoice“of“estimator‘zfor“bMÞoth“the“ph˜ysical“moMÞdel“and“the“excess“struc-Ž¡‘à
ture.‘œNThe›ŒBa•²!y“esian˜p•MÞosterior˜mean,‘Ëmo“de,‘Ëand˜median,–Ëfor˜example,“are‘‹not˜the˜same.Ž¡‘à
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ux“v‘ÿdDaries˜among“the“binsŽ¡‘à
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ux.‘ê»A˜t“the‘ª¯same“time,‘«Ãthe“t˜w˜o“tough“questions,‘«ÂÊQuestions“¹3“and“4,Ž¡‘à
require–¦fmore“complex“treatmen²!t.ŽŽŽŒ‹ H U«ã ý¯t¹’V™ÌP²!oisson‘¦fGo•MÞo“dness-of-t‘HE¹3ŽŽ >z ýÞ† ‘à
ÊPršÁ´oblem–"ù3:‘jMul‘ÿEtiple“Bins,“Ga˜ussian“St–ÿEa“tistics.Ž‘à
Ÿº¥‰ fe|DŽ’&O†¹Presume–ïîone“has“a“spMÞectral“en-Ž¤=k‘à
ergy–¢†distribution›¢‡or“image,‘£Mwith“indepMÞenden²!t˜Gaussian“measuremen²!ts˜and“heteroscedasticŽ¡‘à
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gon›£and‘¢Bo•²!wy“er˜(1976)–¢and˜Cash“(1976)˜suggest,‘ˆüone˜nds“the˜moMÞde“(or˜moMÞdes)“b²!y˜maxi-Ž¡‘à
mizing›m`the‘malog-lik²!eliho•MÞo“d˜function˜via–maone˜of˜a“n•²!um“bMÞer˜of˜standard‘man“umerical˜optimizationŽ¡‘à
methoMÞds–¾2(as›¾3in“Press“et˜al“1992).‘%BThe˜region“around“the˜moMÞde“can“then˜bšMÞe“mapp˜ed.‘%BTheŽ¡‘à
corresp•MÞondence›a^b“et•²!w“een˜drop˜in˜log-lik“eliho•MÞo“d˜and‘a_in•²!terv‘ÿdDal˜co“v“erage˜is˜assumed˜(via˜theŽ¡‘à
Cen²!tral–2Limit›2
Theorem)“to˜bMÞe“giv•²!en˜b“y–2¾Ÿü¾¼2ŽŸ®Aó×2 cmmi8¿ŽŽ‘ eœ¹regardless“of˜the“details˜of“the˜moMÞdel.‘€ËAlter-Ž¡‘à
nativ•²!ely‘ÿe,‘”a›d‰Ba“y“esian˜pro•MÞcedure˜maximizes˜and˜maps˜out˜the˜p“osterior˜probabilit²!y˜of˜theŽ¡‘à
parameters–Ígiv²!en›Ìthe“moMÞdel˜and“data.‘8The“probabilit²!y˜that“a˜parameter“is˜within“a˜cer-Ž¡‘à
tain–]ÿrange“(\credible“region")“is“then“calculated‘^ directly“(as“in“Loredo“1992“and“referencesŽ¡‘à
therein).Ž¦‘à
ÊQuestion‘ÈV2:‘1zÄEvidenc‘ÿp¹e–{Ífor›{Îan“additional˜c–ÿp¹omp“onent˜of‘{Íknown˜shap“e.‘¶U¹Since–Cãa“paramet-Ž¡‘à
ric–û¯function“for“the“pšMÞossible“additional“comp˜onenš²!t“is‘û°kno˜wn,‘one“builds“on“the“proMÞcedureŽ¡‘à
describ•MÞed›”Xab“o•²!v“e˜for‘”Yone˜bin˜in˜t“w“o˜w“a“ys.‘§´First,‘ÏÔone˜in“v“estigates˜the˜limits‘”Yon˜the˜totalŽ¡‘à
ux–{(or“equiv‘ÿdDalenš²!t)‘{ of“the“unkno˜wn“compMÞonen˜t“b˜y“computing“condence‘{ or“credible“in˜ter-Ž¡‘à
v‘ÿdDals.‘‡Second,‘öCone›æIin•²!v“estigates–æJwhether˜the“rise˜in“log-lik²!eliho•MÞo“d˜up“on–æJthe˜addition“of˜thisŽ¡‘à
compMÞonen²!t–âsignal›áit“is“statistically˜signican²!t.‘,Q(W‘ÿee˜remind“astroph²!ysicists“that˜for“man²!yŽ¡‘à
applications,‘Àoften–¿{including“pšMÞoin²!t“sources“and‘¿zsp˜ectral“lines,‘Àthe“mapping“b˜et•²!w“een‘¿{theŽ¡‘à
rise›;¹in‘;ºlog-lik²!eliho•MÞo“d˜and–;ºthe˜c²!hance“signicance˜cannot˜bMÞe“compared˜to“a˜pre-computedŽ¡‘à
reference–Ñšdistribution›Ñ›suc²!h“as˜¾Ÿü¾¼2Ž‘À¹,‘gbut“m²!ust˜bMÞe“calibrated˜via“appropriate˜Mon²!te“Carlo;Ž¡‘à
Protasso²!v–¦fet“al.‘ÝÝ2002)Ž¦‘à
ÊQuestion‘…I3:‘1zÄIs–`¿the“b›ÿp¹est‘`Àt“go˜o˜d“enough?‘«Î¹One–8can›7compute“a˜¾Ÿü¾¼2Ž‘Ð<¹on˜¾‘ª¹degrees˜of“freedom,Ž¡‘à
and–-Þcompare“it“to“a“previously“tabulated“\reference“distribution"“(i.e.,‘Eù¾Ÿü¾¼2ŽŸ®A¿ŽŽ‘3’¹)“bš²!y“in˜v˜oking“theŽ¡‘à
Cen²!tral–‚ILimit›‚JTheorem“(e.g.,‘¹BEadie˜et“al.‘q‡1971˜and“Bevington˜1969).‘q‡There˜is“nothingŽ¡‘à
tec²!hnically–-eincorrect‘-dabšMÞout“this“simple,‘O$p˜opular“pro˜cedure.‘rÙHo•²!w“ev“er–-e¾Ÿü¾¼2Ž‘íi¹is‘-dalmost“alw•²!a“ysŽ¡‘à
an–,ÿappro²!ximation›- to“the˜true“distribution.‘µfHence“an²!y˜statistic“that“relies˜on“its˜tail“(whereŽ¡‘à
the–_®dierences“w²!ould“bšMÞe“the“most“pronounced)“should‘_b˜e“viewš²!ed“with“sk˜epticism.‘ µAlso,Ž¡‘à
Loredo–(1992)›pMÞoin²!ts“out“that˜oMÞccasionally“one“should˜Äexp–ÿp¹e“ct–¹data“that˜are“not“a˜\go•MÞo“d‘t"Ž¡‘à
to–¦fone's“moMÞdel,“bš²!y“c˜hance.Ž¦‘à
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ical–*µmoMÞdel›*¶is“not“sucien²!t˜to“describMÞe˜the“data,‘KÉthe“dierence˜bMÞet•²!w“een–*µthe“data˜and“theŽ¡‘à
ph²!ysical›ZmoMÞdel,–,i.e.,“the˜residual,“is˜a˜reasonable–Yestimate˜of˜additional“unkno²!wn˜compMÞo-Ž¡‘à
nen•²!t.‘ÆBUncertain“ties›_—can–_–bMÞe“computed“b²!y“appropriately“scaling˜the“pixel-wise“observ‘ÿdDationalŽ¡‘à
v‘ÿdDariances.‘¸nIf–6something›6is“kno²!wn˜abMÞout“the˜structure“of“the˜additional“compMÞonen•²!t˜(suc“h‘6asŽ¡‘à
lo•MÞcal›&smo“othness)˜one˜could˜gain˜additional˜p“o•²!w“er˜b“y˜using˜a˜moMÞdel˜that˜formalizes˜thisŽ¡‘à
kno•²!wledge.‘Ü£Ho“w“ev“er›¢ºthere–¢¹is“nothing“actually˜wrong“with“simply“subtracting˜the“ph²!ysicalŽ¡‘à
moMÞdel–¦ffrom“the“data,“as“is“usually“done.ŽŽŽŒ‹ Þ U«ã ý¯t¹’V™ÌP²!oisson‘¦fGo•MÞo“dness-of-t‘HE¹4ŽŽ >z ýÞ† ‘à
ÊPrÁ´oblem–"ù4:‘jMul‘ÿEtiple“Bins,“Poisson“Counts.Ž‘à
Ÿº¥‰ fe òL玒 )¹This–¦ºis“the“same‘¦¹as“ÊPrÁ´oblem‘#Q3¹,‘¦ÏbutŽ¤=k‘à
with–¦fPš²!oisson“data,“i.e.,“discrete“coun˜ts.Ž©£‘à
ÊQuestion‘@´1:‘1zÄPar–ÿp¹ametrize“d›öphysic“al˜mo“del|b“est–÷t˜and“unc‘ÿp¹ertainties–¶¹and“ÊQuestion‘@´2:Ž¡‘à
ÄEvidenc‘ÿp¹e–·qfor›·ran“additional˜c–ÿp¹omp“onent–·qof“known˜shap‘ÿp¹e¹.‘Ë7The–nsproMÞcedures›ntfor“these˜questionsŽ¡‘à
are–¦fessenš²!tially“the“same“as“for“the“Gaussian“case“(Cash“1979;“Protasso˜v“et“al.‘ÝÝ2002).Ž¦‘à
ÊQuestion‘0¨3:‘1zÄIs–×5the›×6b‘ÿp¹est“t˜go–ÿp¹o“d‘×5enough?›àe¹Here–§>is“our“c²!hallenge.˜In“the“P²!oisson“case,Ž¡‘à
there–“is›’no“pre-computed,–5½single,“reference˜distribution–“that“doMÞes˜not“depMÞend“on˜the“detailsŽ¡‘à
of–¦fthe“moMÞdel“and“data.Ž¦‘à
ÊQuestion‘>¤4:‘1zÄQuantify–ãvunknown“additional“structur›ÿp¹e“and“its‘ãuunc˜ertainties.‘Y¹If‘´one‘´hasŽ¡‘à
no›|ègo•MÞo“d˜prior˜kno²!wledge˜of˜a˜simple‘|éparametric˜mo“del˜for˜the˜\mismatc²!h"˜b“et•²!w“een˜theŽ¡‘à
data–ÒŒand“phš²!ysical“moMÞdel,‘Ý–ho˜w“došMÞes“one“estimate“shap˜e‘Òand“signicance“of“the“mismatc²!h?Ž¡‘à
Subtracting–´¼the›´½moMÞdel“from“the˜data“is“no“longer˜a“handy“estimator˜ev²!en“of“the˜mean.‘OEv²!enŽ¡‘à
if–¡dthe“counš²!ts‘¡eare“relativ˜ely“high,‘¢dwhen‘¡eone“is“in˜terested“in“lo˜w‘¡eprobabilit˜y“ev˜en˜ts|suc˜h“asŽ¡‘à
the–¤counš²!ts“from‘¤¬the“unkno˜wn“additional“structure|it‘¤¬is“impMÞortan˜t“to“tak˜e‘¤¬full“accoun˜t“ofŽ¡‘à
the–¦fskš²!ew“and“long“tail“of“the“underlying“P˜oisson“distribution.ŽŸ¶‘)ÆwW‘ÿee–o©brie
y“note“that‘oªalthough“wš²!e“ha˜v˜e“not“discussed“instrumen˜tal‘oª\blurring"“or“\smear-Ž¡‘à
ing",‘$)if–the›blurring“matrix˜or“pMÞoin²!t˜spread˜function“is˜assumed“to˜bMÞe“kno²!wn˜completely‘ÿe,Ž¡‘à
it›is‘straigh•²!tforw“ard˜to˜con“v“olv“e–it˜with˜the˜sky“and˜ph²!ysical˜mošMÞdels“b˜efore‘comparisonŽ¡‘à
with–Úthe“data.›:Th•²!us,‘&ww“e–Úmainly“ignore“these“issues“throughout“this“article.˜(The“case“ofŽ¡‘à
uncertain•²!t“y–P9in“instrumen²!tal‘P8respšMÞonse“is“b˜eing“addressed‘P8in“a“hierarcš²!hical“w˜a˜y‘P8in“a“separateŽ¡‘à
eort;–¦fsee“Drak²!e“et“al.‘ÝÝ2006.)ŽŸ ƒ7‘à
Å2.Ž‘*C High–2Energy“Gamma-Raš¦ty“Bridge“Ov˜er“the“Milky“W–þó\a˜y“,–2circa“1999ŽŸ[‘à
2.1.Ž‘4JBrief–hGamma-Raš¦ty“Astronom˜y“Ov˜erview:‘*Matter‘g&“Energy“in“our“GalaxyŽŸå¿‘à
¹Gamma-ra²!y–ŠAemission“(and›ŠBhence“detection)“is“an“instrinsically˜P²!oisson“proMÞcess“(LongairŽ¡‘à
1992).‘×çA•²!t›”‚v“ery˜high›”ƒenergies,‘˜ev“en˜after–”‚almost˜a“decade˜of“high-resolution˜satellite“obser-Ž¡‘à
v‘ÿdDations–äFare›äEsummed,‘ó¾there“are˜man²!y“regions“in˜the“sky“with“v•²!ery˜lo“w›äFcoun“ts˜pMÞer‘äEbin˜(seeŽ¡‘à
Figure–¦f1a,“Left).‘ÝÝGaussian“appro²!ximations“are“not“suitable.Ž©¶‘)ÆwDetailed–}Jphš²!ysics“theory“|“suc˜h“as‘}Iquan˜tum“mec˜hanics“|“allo˜ws“researc˜hers“to“`decoMÞde'Ž¡‘à
the–Ðbasic›Ïph²!ysics“of˜distan²!t“ob‘ ›»jects“via˜their“remotely-detected˜photons.‘ÖVAcross˜the“en²!tireŽ¡‘à
electro-magnetic–·&spMÞectrum,›»Vfrom“radio“(lo•²!w“est–·&energies)‘·%through“infrared,˜optical,˜UV,“X-Ž¡‘à
ra•²!y‘ÿe,‘Nôand›9gamma-ra“ys–9(highest˜energies),‘Nôphotons˜tell“us˜ab•MÞout˜sp“ecic‘9pro“cesses˜in‘9the˜sky‘ÿe.Ž¡‘à
Gamma-ra²!y–tBemission“originates,‘~Jfor“example,‘~Ifrom“high-energy‘tCcosmic“proMÞcesses“includingŽ¡‘à
n•²!uclear›ÄZdeca“y˜and–Ä[high˜energy˜particles˜impinging“on˜the˜magnetic˜elds,‘ËØdense˜material,Ž¡‘à
or›lo•²!w“er‘¬energy˜photons.‘û±In˜particular,‘~gamma-ra“ys‘¬rev“eal˜the˜whereabMÞouts‘¬of˜extremelyŽ¡‘à
energetic–¦fparticles,“in“our“Milky“W‘ÿeaš²!y“Galaxy“and“bMÞey˜ond.Ž¦‘)ÆwGivš²!en–þàthis,‘ bdo“w˜e›þáunderstand“the˜proMÞduction“cycle˜of“suc²!h˜highly“energetic˜matter“andŽ¡‘à
energy–ö’in“our“Galaxy?›Î`Or“is“there“more“\stu‘ Ú"“w²!e“are“missing?˜F‘ÿerom“our“v‘ÿdDanš²!tage“pMÞoin˜tŽŽŽŒ‹ 1ª U«ã ý¯t¹’V™ÌP²!oisson‘¦fGo•MÞo“dness-of-t‘HE¹5ŽŽ >z þxӟǨ‘;š<ï]PSfile="cgroegretcountsg1t9g004loresgs2.eps" llx=36 lly=82 urx=325 ury=278 rwi=1728 rhi=1080 Ž’ ì³Øï]PSfile="cgroegretexposrg1t9g004loresgs2.eps" llx=36 lly=82 urx=325 ury=278 rwi=1728 rhi=1080 ŽŽŽŸÌÈ‘)Æw²Figure‘¨1.‘ CGR¸ãO/EGRET‘–data:–Pqa)›§Left:“µ>²1˜GeV‘—coun•¸ãts‘¨measured˜o“v“er–¨the˜whole“skyŽ¤
ÌÈ‘)Æwo•¸ãv“er–xËthe›xÊlifetime“of“the˜mission“(from“HEASAR¸ãC),˜plotted“in“Galactic˜coGordinates“inŽ¡‘)Æw0µ:²5Ÿü^ÿóO!â… cmsy7·Ž‘ÑÄó
!",š
cmsy10¸‘8à²0µ:²5Ÿü^ÿ·Ž‘î9²bins.–qÇb)›UURigh¸ãt:“corresp•Gonding˜exp“osure˜(eectiv¸ãe˜area˜times˜time).ŽŽŸ ‘à
¹t•²!w“o–ÔRthirds“of“the“w•²!a“y‘ÔQout–ÔRfrom“the“cenš²!ter“of“the“Milky“W‘ÿea˜y“Galaxy‘ÿe,‘ßÌho˜w“w˜ell“can“w˜e“useŽ¤=k‘à
the–¦finformation“pac•²!k“ed–¦fin“gamma-ra²!y“photons“to“learn“abšMÞout“these“pro˜cesses?ŽŸ[‘à
Å2.2.Ž‘4JThe–2Gamma-Ra¦ty“SkyŽŸå¿‘à
Instrumen¦tts.‘
ó5¹These–0Îw²!ere›0Íthe“questions˜and“tec•²!hniques˜w“e–0Îfaced˜in“the˜1990's,‘HRwhen“theŽ¡‘à
ÄCompton‘[Gamma-R‘ÿp¹ay‘[ Observatory– ]¹(CGR²!O),“the› ^rst“of“NASA's˜\Great“Observ‘ÿdDatories",Ž¡‘à
surv•²!ey“ed–Ñthe“enš²!tire“sky“for“gamma-ra˜y‘Ñemission.‘]íCGR˜O‘ÐÉcarried“t˜w˜o“surv˜ey“telescopMÞes:Ž¡‘à
COMPTEL‘‘¿(0.5-30–‘üMeV,›‘ûthe“n²!uclear“regime);‘Æand˜EGRET‘‘À(30˜MeV-100“GeV).˜It“alsoŽ¡‘à
carried›Ÿ©t•²!w“o‘Ÿªnon-imaging˜instrumen“ts:‘ÐcBA‘ÿeTSE,‘Ÿªwhic“h˜foMÞcused˜on˜gathering‘Ÿªsignals˜fromŽ¡‘à
the–E¶whole“sky›Eµas“a“function“of“time“(in˜order“to“observš²!e“gamma-ra˜y“bursts);‘•]and“OSSE,Ž¡‘à
whic•²!h›¦fconcen“trated˜on˜spMÞectra.Ž©¶‘)ÆwOf–,Ãthese,‘NYthe“instrumenš²!tal“respMÞonse“of‘,ÂCGR˜O-EGRET's“Spark“Cham˜bMÞer“is‘,Âthe“sim-Ž¡‘à
plest,‘XŸand›4ûits–4úparticle“induced˜bac²!kground“is˜the“lo•²!w“est.‘‰šHence˜for–4úthis“papMÞer,‘X where“w²!eŽ¡‘à
ignore›åm\decon•²!v“olution",‘5/w“e˜used–ån¾>‘[¹1GeV‘åEGRET‘åSpark“Cham²!bMÞer˜data˜(real“and˜sim-Ž¡‘à
ulated),–#binned‘×_in²!to›×^1¾:¹5Ÿü¾ó¾KÈ cmsy8ÂŽ‘ú@ó!",š
ó3
cmsy10Á‘:<¹2Ÿü¾ÂŽ‘ —c¹bins,“a˜levš²!el–×_of“resolution“where“instrumen˜tal“blurring“isŽ¡‘à
minimal.ŽŸ£‘à
ÅCGR¦tO/EGRET‘5bdata.‘
ó5¹In‘‡´Figure–‡³1a“(on“the“left),‘Àwš²!e“sho˜w“the“¾>¹1‘‡´GeV‘‡ycoun˜ts“(pMÞerŽ¡‘à
0¾:¹5Ÿü¾ÂŽ‘s
Á‘³¹0¾:¹5Ÿü¾ÂŽ‘Ì’¹bin)–Žcollected“bš²!y“the–CGR˜O/EGRET‘stelescopMÞe“during–Žits“Á¹9+“y˜ears“viewingŽ¡‘à
the–"sky“(Nolan›!et“al.‘’1992,‘ÆÐThompson“et“al.‘’1993,‘ÆÑEspMÞosito“et˜al.‘’1999,‘ÆÐMatto²!x“et“al.Ž¡‘à
1992).‘Ö?On–the›Œrigh²!t“is“the˜correspšMÞonding“EGRET‘‡telescop˜e›Œeectiv²!e“area,‘”in“units˜of“cmŽ‘lŸü¾¼2Ž‘Mp¹.Ž¡‘à
Visible–&§in›&¨the“plot˜of“gamma-ra•²!y˜coun“ts–&§are˜a“couple˜hš²!undred“\pMÞoin˜t‘&¨sources"“(HartmanŽ¡‘à
et›_,al.–Æ1999).“A‘_n•²!um“bMÞer˜are˜though“t˜to˜bMÞe˜spatially˜unresolv“ed˜glo“wing˜gas˜from˜supMÞerno“v‘ÿdDaŽ¡‘à
remnan²!ts–ë«(SNRs)›ëª(EspMÞosito“et“al.–«1996).“Virtually˜all–ë«of“the˜others“are“understo•MÞo“d˜to‘ë«b“eŽ¡‘à
compact–‚*ob‘ ›»jects“sucš²!h“as“neutron“stars“and“accreting“blac˜k“holes.‘q)These“include“pulsarsŽ¡‘à
(neutron–_Ëstars“with“magnetic›_Êelds“roughly“a“dozen“orders“of˜magnitude“greater“than“thatŽ¡‘à
of–¦fthe“Earth)“and“v‘ÿdDarious“kinds“of“activ²!e“galaxies.Ž¦‘)ÆwThe–9×bulk›9Øof“the“signal“in˜this“lo•²!w-bac“kground,‘Olo“w˜coun“t›9×regime,‘Oho“w“ev“er,‘OŽcomes˜fromŽ¡‘à
diuse–ʈgas“and‘ʇnearbš²!y“bac˜kground“photons“along›Ê‡the“plane“of“our˜galaxy“(e.g.,‘ÓCillis“andŽŽŽŒ‹ A½ U«ã ý¯t¹’V™ÌP²!oisson‘¦fGo•MÞo“dness-of-t‘HE¹6ŽŽ >z ýÞ† ‘à
Hartman–¸—2005,›½$Elw²!e“2003,˜and‘¸˜references“therein),˜glo²!wing“as“they›¸˜are“impacted˜b²!y“ener-Ž¤=k‘à
getic–
particles“called“cosmic“ra•²!ys.‘ÀThere‘
w“ere›
ph“ysics-based˜moMÞdels˜a“v‘ÿdDailable˜at˜the˜timeŽ¡‘à
that–?predicted›?what“the“diuse˜glo²!w“should˜loMÞok“likš²!e“(Bertsc˜h‘?et“al.‘§ã1993;‘‹iHun˜ter‘?et“al.Ž¡‘à
1997,›aStrong‘Þaand–Þ`Mosk‘ÿdDalenk²!o“1998,˜Mosk‘ÿdDalenk²!o,‘bStrong,˜and“Reimer“1998,‘bMosk‘ÿdDalenk²!o“andŽ¡‘à
Strong–2 1998).‘·EThese“mošMÞdels“of“the“Galaxy‘2Ÿincluded“sup˜erno²!v‘ÿdDae“as“sources‘2Ÿof“energetic“cos-Ž¡‘à
mic–ݽraš²!y“particles;‘ the“spatially“complex“gamma-ra˜y“glo˜w‘ݼfrom“energetic“particles“impingingŽ¡‘à
on–5Âgas›5Áclouds“(Bremmstrahlung˜and“Pion“deca²!y);‘[Nand˜a“broader,‘LIsmo•MÞother˜comp“onen²!t‘5ÂdueŽ¡‘à
to–Ò3these›Ò2energetic“particles“\b•MÞo“osting"˜the–Ò3energy“of“photons˜from“the“ubiquitous˜lo²!w“en-Ž¡‘à
ergy›Œ(micro•²!w“a“v“e˜and˜infrared)˜bac“kground˜(In“v“erse˜Compton).‘NThe˜compMÞonen“ts˜of˜theseŽ¡‘à
phš²!ysical–DÏmoMÞdels“w˜ere“measured“separately‘ÿe.‘¹High“resolution“radio“observ‘ÿdDations“w˜ere“usedŽ¡‘à
to–lêmap›lëout“the“gas˜distribution,‘xiand˜loMÞcal“measuremen²!ts“of˜cosmic“ra²!y“particles˜w²!ere“usedŽ¡‘à
to–¦fconstrain“their“energy“spMÞectra“and“abundances.ŽŸ¶‘)ÆwWith–‹Bthe‘‹Cphš²!ysical“moMÞdel“and“CGR˜O‘‹observ‘ÿdDations“in“place,‘Äya‘‹Cfundamen˜tal“questionŽ¡‘à
wš²!as–¦fwhether“the“moMÞdels“matc˜hed“the“observ˜ed“data.ŽŸõŽ‘à
Å2.3.Ž‘4JImaging–2MethoY‹ds“for“the“Whole“Sky“in“Gamma-Ra¦tysŽŸå¿‘à
CGR¦tO‘÷‘era.‘
ó5¹In–Qäorder“to‘Qãansw²!er“this“question,‘|ÃDixon“et“al.‘àV(1999)“applied“the“TIPSHŽ¡‘à
(translation›upin•²!v‘ÿdDarian“t˜P“oisson˜smoMÞothing˜using˜Haar‘uqw“a“v“elets)˜methoMÞd˜of˜Kolaczyk˜(1997)Ž¡‘à
and– Dixon“and“Kolaczyk› Œ(2000).‘ÛêThat“is,‘¡¹in˜ÊQuestion“¹4“under“sc“Problem“4“abMÞo•²!v“e,‘¡¸theyŽ¡‘à
mošMÞdeled–the‘\mismatc²!h"“b˜et•²!w“een›the‘ph“ysical˜moMÞdel–and˜the“CGR²!O/EGRET‘òall-sky“dataŽ¡‘à
with–Ôia“mš²!ultiscale“(b•MÞo˜xy-shap“ed–ÔiHaar“w˜a˜v˜elet,‘êwith“cycle-spinning)“moMÞdel.‘gæThe“TIPSHŽ¡‘à
\kš²!eep–‘Ÿor“kill"“thresholds“w˜ere“carefully“tailored“to“matc˜h“the“P˜oisson“statistics.‘ŸˆA˜t“theŽ¡‘à
time,‘ýthey–¸Œdid›¸nd“a˜mismatc²!h“that˜loMÞok²!ed˜rather“lik²!e˜a“bridge˜abMÞo•²!v“e–¸Œthe˜plane“of˜theŽ¡‘à
Galaxy‘ÿe.‘ÝÝThey–¦ffamously“emphasized,“ho•²!w“ev“er:ŽŸëÄ‘/@\The–Wéimmediate“question“arises“as“to“the“statistical“signicance“of“this“feature.Ž¡‘/@Though–GÜw²!e›GÝare“able“to˜makš²!e“rigorous“statemen˜ts“abšMÞout‘GÝthe“co˜ecien²!t-wiseŽ¡‘/@and–ùûlevš²!el-wise‘ùüFDR‘ùÐ[F‘ÿealse“Disco˜v˜ery‘ùüRate],‘wsimilar“quan˜tication‘ùüof“ob‘ ›»ject-wiseŽ¡‘/@signicance–¦f(e.g.,“`this“blob“is“signicanš²!t“at“the“¾n“¹sigma“lev˜el')“are“dicult."ŽŸëÅ‘)ÆwThe–1|use“of‘1{mš²!ulti-scale“methoMÞds“w˜as“a“notable“impro˜v˜emen˜t‘1{in“moMÞdeling“unkno˜wnŽ¡‘à
diuse–|`emission,‘„Çwhicš²!h“is“expMÞected“to“sho˜w›|_bMÞoth“the“sharp“and“broad“features˜of“the“com-Ž¡‘à
plicated–û[diuse›ûZgas,‘P—the“photon˜eld,‘P—and“sea˜of“energetic˜particles“(cosmic˜ra²!ys)“fromŽ¡‘à
whicš²!h–:it“arises.‘0ZPreviously‘ÿe,‘3oamong“astroph˜ysicists,‘3oother“prominen˜t‘;\moMÞdel-free"“or“non-Ž¡‘à
parametric–;'(i.e.,‘`Wa“mošMÞdel“with“ab˜out“as‘;&man²!y“parameters“as“data“bins)“metho˜ds“includeŽ¡‘à
Ric²!hardson-Lucy–Â(the›ÂEM‘Âalgorithm“applied˜to“an˜unconstrained“grid“of˜pMÞoisson“parame-Ž¡‘à
ters;‘ÞÎRic²!hardson–Ì1972,›ÕhLucy“1974,˜Shepp“and“V‘ÿeardi“1982)“and“Maxim•²!um-En“trop“y-relatedŽ¡‘à
methoMÞds–}(sucš²!h“as“MaxEn˜t“and‘~successors;›E Skilling“1989;˜also‘~Strong“2003).‘:These“methoMÞdsŽ¡‘à
lac•²!k“ed–¦fthe“mš²!ultiscale“represen˜tation“of“bMÞoth“diuse“and“sharp“features.ŽŸèÖ‘à
ÅProgress‘ëù1:‘uqNK.‘
ó5¹In›ithe–iensuing“y•²!ears,‘uMsev“eral‘ipMÞoten“tial˜impro“v“emen“ts›iw“ere˜tried.‘Éh(SeeŽ¡‘à
also–M¯Willett›M®2006“and˜Y‘ÿeoung“2006,‘_lthis“v²!olume;‘kAand“the˜dierenš²!t“approac˜hes‘M®of,–_msa˜y‘ÿe,“Bour-Ž¡‘à
din–õ»et“al.‘¢ù2001;‘0ŸStarc•²!k,›P“an“tin‘õºand–õ»Murtagh“2002,˜Starc²!k‘õºand“Murtagh“2006“and“referencesŽ¡‘à
therein.)‘£õThe›øTIPSH‘ø€w•²!a“v“elet˜structure˜w“as‘ø¬rev‘ÿdDampMÞed˜in“to˜a˜more‘ø¬general˜m“ultiscale˜struc-Ž¡‘à
ture–I•on›I–certain“ratios“of“expMÞected˜coun²!ts“to“exploit˜the“statistical“propMÞerties“of˜the“P²!oissonŽŽŽŒ‹ P U«ã ý¯t¹’V™ÌP²!oisson‘¦fGo•MÞo“dness-of-t‘HE¹7ŽŽ >z ýÞ† ‘à
likš²!eliho•MÞo“d–;Band‘;Ato“allo˜w›;Athe“inclusion˜of“instrumen²!t˜respMÞonse“(\smearing"˜or“blurring)˜in“anŽ¤=k‘à
eleganš²!t–Z£(and“spMÞeedy)“w˜a˜y“(No˜w˜ak“and‘Z¢Kolaczyk“2000;‘´ÁKolaczyk“and“No˜w˜ak“2004,‘‡²2005;Ž¡‘à
henceforth–þIthis“methoMÞd“is“referred“to“as“NK).“Likš²!e“w˜a˜v˜elets“(whic˜h“often“use“recursiv˜eŽ¡‘à
dyš²!adic–4³partitioning),‘XFthe“NK‘4structure“breaks“an“image“in˜to“four“quadran˜ts“(or“t˜w˜o“seg-Ž¡‘à
menš²!ts,‘kÇin–]1D)‘]
and‘] then“successiv˜ely“breaks‘] eac˜h“of“these‘] quadran˜ts“in˜to“its‘] four“quadran˜tsŽ¡‘à
unš²!til–åÇthe“inheren˜t“pixelization“is‘åÈreac˜hed.‘œ Unlik˜e“w˜a˜v˜elets,‘õŸfor‘åÈwhic˜h“the“basis“function“isŽ¡‘à
t²!ypically–•‹some›•Šsort“of“dierencing“function,‘˜éfor“NK‘•‡the˜basic“shapMÞe“is“a˜constanš²!t“in˜tensit˜yŽ¡‘à
bloMÞcš²!k.‘çóF‘ÿeurther,‘ªšrather–©Ãthan“w˜orking‘©Âwith“the“in˜tensit˜y‘©Âitself,‘ªšw˜e“consider“the“ratios‘©Âof“theŽ¡‘à
inš²!tensities–õof“eac˜h“quadran˜t“to‘öthe“whole“(see“Willet“2006,‘6this“v˜olume).‘¯
These“\split“prob-Ž¡‘à
abilities"–ÕŒare›ÕdescribMÞed“b²!y˜prior“distributions˜tuned“b²!y˜smoMÞothing“parameters,‘áWin“analogyŽ¡‘à
to–Y¢the“w•²!a“y›Y£that–Y¢shrink‘ÿdDage“(or“complexit²!y“pMÞenalties)˜are“used“to“estimate“the˜bMÞest“v‘ÿdDalues“ofŽ¡‘à
coMÞecienš²!ts–Äin‘Äa“w˜a˜v˜elet“decompMÞosition.‘6èF‘ÿeurther,‘Ë€NK‘Ä
is“form˜ulated›Äso“that“the˜problem“isŽ¡‘à
con•²!v“ex–¶and“there“is‘¶only“one“mošMÞde,‘¹ïso“go˜o˜d‘¶con•²!v“ergence–¶is“assured.‘ÀFinally‘ÿe,‘¹ðas“with“mostŽ¡‘à
w•²!a“v“elet–¢manalyses,‘áocycle-spinning“is›¢nrecommended“to“remo•²!v“e˜artifacts–¢min“the˜tted“imageŽ¡‘à
(Coifman–¦tand“Donoho,‘¦x1995).‘ÞStill,‘¦wthis“used“the“EM“algorithm“to“get“a“\bMÞest“t"“image,Ž¡‘à
and›¦funcertain•²!t“y˜estimates˜w“ere˜not˜a“v‘ÿdDailable.ŽŸwñ‘à
ÅProgress‘ÃH2:‘º2EMC2.‘
ó5¹Later,‘YûEscš²!h–Ýet“al.‘óC(2004)“em˜bMÞedded“the“NK‘„m˜ultiscale“moMÞdelŽ¡‘à
in•²!to›‘ûa‘‘úhierac“hical˜Ba“y“esian˜framew“ork.‘ ›(See˜Karo“vsk‘ÿdDa˜et–‘úal.,‘Ìà2005˜for“applications˜withŽ¡‘à
bMÞeautiful–D°graphics.)‘½KThis›D¯also“allo²!ws“the˜smoMÞothing“parameters“used“in˜NK‘D—to“bMÞe“t˜to“theŽ¡‘à
data.‘¦The–é"range›é!of“v‘ÿdDariation“of“the“tuning“parameters“is˜eectivš²!ely“go˜v˜erned“b˜y“a“h˜ypMÞer-Ž¡‘à
parameter–Lthat›Lis“set“ahead“of˜time.‘¿ÁIn“order“to“fully˜map“out“the“pMÞosterior˜distribution“ofŽ¡‘à
the–¸«v‘ÿdDarious›¸ªmoMÞdel“and“tuning“parameters,‘½