Bayesian derivation of LRs for continuous variables
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1 Bes ervo o s or coos vrbles Abel Bolck * Ivo Alberk eerls Foresc Ise P. O. Bo AA Te ge Te eerls. *corresog or :.bolck@.js.l eerls Foresc Ise P. O. Bo AA Te ge Te eerls. Teleoe: F: Absrc. A esre or e sreg o evece oresc corso cses were es re core o vesge weer e orge ro e se sorce s e lkeloo ro. Te lerre roves orls or clclg s bse o e vles o e crcerscs o be core. Ts er gves lerve orls or coos crcerscs erve sg sr Bes eor. clclo s eee or sos were e vrces w bces re o sse eql. Kewors: Bes eor evece evlo oresc corso lkeloo ro lvre crcerscs. Iroco I oresc scece reqel corsos re erore o wo or ore gros o es o evece e.g. soe o soe-r or cre scee glss rcle w rcle ssec s r. Te qeso o eres e s weer e es orge ro coo sorce. Ts s lso e cse we e coo sorce s o kow sc s w bles cog MDMA []. Two sceros re cosere sll o e or : es A B ve coo sorce : es A B re ro ere
2 sorces. Te oeses re reerre o s rosecor s eece oess. Te sreg o evece o e gs o corsos or eer o e oeses s og o be bes qe b e so-clle Lkeloo Ro. I s ee s e ro o e robbl es o s evece gve e wo oeses. I oresc corsos e evece oe cosss o wo rs. O e oe crcerscs ossbl reee o e cre scee e o e oer ose o e ssec e. Eve o cler sco bewee cre scee ssec e c be e sc s w MDMA bles were oe s ol erese weer e coe ro e se bc oe se o es or bc c be reerre o s corol e oer s qesoe es or bc o core w s corol. I e er we cocere o e so wc e crcerscs beg esre re coos lvre. Focs wll be o ervo o orls or e s sg sr Bes resls gve orl ssos o e e esrees orl or kerel es esos KDE o e erlg olo. Te clsscl resl or coos crcerscs c be o Lle [] wc covers e cse o vre w eql vrce o e esree errors. Ake Lc [3] cover e cse o lvre lso llowg kerel es eso o e robbl srbo o e crcerscs cosere over e olo. ere s well covrce rces o e esree errors re cosere eql. I s er we se o o gve lerve w o obg e orls c be o Ake Lc [3] b coog o oe o e ses o es sg sr Bes resls seco. Tog ecll eqvle e orls resee ere look rer ere. Tese orls c esl be erve rove. Fll e orlo llows e vrce o e seqeces o esrees o e wo gros o es o be ere. Ts s or ele sble or corso o MDMA bles becse e vrce o crcerscs w bc vr rcll ro bc o bc. Beses s cre scee sles ve c lrger vro ssec sles eve orgg ro e se sorce or ele becse o ollo o e rce. I g e roc s worwle ro ccl o o vew.
3 3. s or Gss crcerscs w Gss ror or e e e.. vre orl crcerscs ro orl olo Te oel s se Ake Lc [3] s wo-level ro eec oel. Ts s o s: we ve esrees rereseg reee esrees o corol e A rereseg esrees o qesoe e B we w o core w e corol e A wc ve orl srbos w e se sr evo es. I e wo-level oel ese e es re sse o be esrees ger level. Te re g ro vrbles orgg ro orl srbo w e sr evo. I orls or ll j: j E j j F j w E j F j were Lle [] Ake Lc [3] sse. B eo o e Lkeloo Ro coog o sg eeece o er e ere e eror e robbl es s gve o gve e srbo o. Te eoor roves e es o gve e olo srbo. sg e wo-level oel gves:. I Bes eor e eror rereses e oseror recve srbo o ew vle er e srbo o e e es s e w oro bo e secc corol e A reresee b e ocoe o. Te eoor rereses e ror recve srbo o ew vle s beore g o e olo
4 4 srbo. Ise o sgle esrees lle verge esrees c be se. Fro ere o le j j j j. Accorg o Alberk Bolck Meges [4] s so cse o orll srbe esree vro w e es ere s o loss o oro oe swces ro cl o verge esrees s. ece ro ere o we wll cocere o verge esrees. sg sr Bes eor cors o Ake Lc [3] e e c be orle s Teore. Teore. I e corol esrees e qesoe esrees re orll srbe w e eql vrces s e ror srbo se s e j j e w. Proo. Accorg o sr Bes eor c. [5] e oseror srbo o e e er e ror srbo s e w e ocoe o s orl w e vrce :
5 5. Te oseror recve srbo o ew vle gve er e eqls Θ Θ. Te ror recve srbo s Θ Θ. ece: φ φ w φ rereseg e es o. Rewre s s e eresso eoe Teore. QED oe: e rs er e eoe o Teore rereses e rr o e qesoe sle e seco er e slr o e qesoe w e corol sle. I e cse e vrces re sse e b o eql so we sse ere sr evos ore geerl or o Teore c be se: e w e vrce relce b e vrce e oseror e vrce b... Mlvre orl crcerscs ro orl olo We r o e cse wc ore crcersc s esre. As e revos seco sose we re lookg esrees wc ve orl srbos w e b o ecessrl eql covrce rces ro es. Te es re g sse o be ro vecors orgg ro orl srbo w e vrce T. I orls: j E j j F j w
6 6 T E j F j or ll j. B eo o e Lkeloo Ro coog o sg eeece o er e g. Teore. I e corol esrees e qesoe esrees re orll srbe w e eql vrces s e ror srbo se s T e ] e[ w. Proo. Accorg o sr Bes eor c. [56] e oseror srbo o e es c be wre s T. Te reer o e roo s slr o e vre cse. QED..3 vre orl crcerscs ro olo reresee b kerel es eso. Oe e es ve rreglr ercl srbos. Te Kerel Des Esos KDE c be se or e olo es. Ts es e ror e vre cse s gve b: e π
7 7 w s os e sle bc es o rg bces j... lso reerre o s e bckgro or reerece. Lke Z. A ol bw or s gve b c. [7] o Teore 3. I e corol esrees e qesoe esrees re orll srbe w e b o ecessrl eql vrces s j j e ror KDE s s bove e: b e e e e w b. Proo. Gve e KDE ror orl esrees w oe gro o es e oseror srbo o er s gve b: C C e e were. C π π
8 8 Te cos C beore e so sgs eror eoor s eql ereore c be reove. Te sg e e s [8] b B A AB c B A b B A w Bb A B A c gves were. sg e c s eee o e oseror e c be rewre s e e e e. A sg geerl e π e e e e π. Te ror recve oseror recve c be erve sg slr rcles. Te ror recve s c e eoor o e oseror srbo bove w relce b. Ts
9 9 e π. Te oseror recve c be erve s. e e e π Ts e wc s e ro o e ror oseror recve srbo s s Teore 3. QED.4 Mlvre orl crcerscs ro olo reresee b kerel es eso. I ore oe crcersc s esre orl o s sse b e ror co be sse orl lvre kerel es esos KDE or e bc es c be sse. T s o s srg ro rg j w bc es we sse e ror s gve b e T T π were T s e covrce r bewee bces. Te b w s ol c. [7] e vle 4 4 o. We ve e ollowg resl: Teore 4. I e corol esrees e qesoe esrees re orll srbe s e ve e KDE ror escrbe bove e
10 T e e e e w. Proo. Slr s e roo o Teore 3 oseror srbo or c be erve: e e e ˆ T T π w. Te slr w s or Teore 3 e orl o Teore 4 c be rove. QED.5 Eso o reg reers Reg reers e.g. eqvles e lvre cse c be ese b M Lkeloo ML esors or blce cses w ore vce ecqes e.g. REML bse o bckgro. A lerve s o work w ll Bes oel w rors or ll reers erreers. Tese er rors sol owever be orve relec e olo srcre oerwse e coce o rr ges los so ese lso ee o be re sg bckgro. A roosl or s s resee [9]. Bo e l [] work w rors or e e vrce or wrg bse o orls lke e oes ro Ake Lc [3]. 4. Dscsso coclso
11 Te orlos o e s s er re erve b sg sr Bes eor cors o Ake Lc [3]. Te bsc rcle o e cosrco o e s s owever e se. A vge o sg Bes eor s e orls c be resee cler ccll eleg w. Also e orls re esl rove. Fll e orls c be eee o cses were e b eql vrces re sse. A rer eeso wll be o se rors or bo e e vrces e e es. Bo e l [] rove MCMC solo bse o Ake Lc s orls. Also orlo sg sr Bes eor c be se. I [9] s sow secl cses close or orls c be erve. 5. Reereces [] Bolck A. Weer C. Djor L. Essev P. v e Berg J. 9 Dere lkeloo ro roces o evle e sreg o evece o MDMA bles corso. Foresc Scece Ierol Vol [] Lle D.V. 977 A roble Foresc Scece Boerk [3] Ake C.G.G. Lc D. 4 Evlo o rce evece e or o lvre. Al. Ss. 53 r 9-. [4] Alberk I. Bolck A. Meges S. Corso o clclo sg verges or rw rero. [5] Gel A. Crl J.B. Ser.S. Rb D.B.4. Bes D Alss seco e. c. 3: C & ll. [6] Aerso T.W. 984 A roco o lvre sscl lss seco e. Jo Wle Sos: ew ork S 84.
12 [7] Slver B. W. 996 Des Eso or Sscs D Alss. C llcrc ew ork S 996. [8] Bo G.E.P To G.C. 99 Bes Ierece Sscl Alss Wle Clsscs Lbrr [9] Alberk I. Bolck A. Meges S. orls sg ll-bes roc rero. [] Bo S. Tro F. Mrqs R. Scbl M. 8 Probblsc evlo o wrg evece; Lkeloo ro or ors. Al. Ss. Pr
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