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N 5!*" 0. .3 -EdW*5* ` 2 !:.Pr en , Ce0؁5!70A!@DB!TZ_-b=S5 ?))1u> 0Y41"!..,=q*2lApproach 1: use two samples55)ҡ4͖.LYA`D C tD&TZ_-byS5 Hª~-"> 0941"!..,u*2H2Assume access to a-` only from the background5L>埊Ȗ˖ UUleyDƦR! 6+D>*DWZb!K*tBUbUn..&Ύ9+- >U4.r~M ŽŽօh_eaoj TZ_A"". . , * aa5Problem: it might not be available in all experimentSZ B o E7 >        Y B WZbM"0...}F'Is204"0...=I*&| 5J+.Fn =6qԎ! oDo',# -W* $ AA^! ѓ  r "0 e [* ;  * NJ$=v$  n H  "decimal)  C 4 *!]E0D* *JG-ZPKJVv{ Index/Slide-3001584.iwa 'URK"  -*4ʢںںںٱ 7d"31B/ Transitionnone?)@?0X*k2Z:}: :::::x UkWXYZ[ >7)-("    @ *֢עȬآ2rr R  PC7}B yDrD%!h%@A-?0(8RWZb`he" K*t8J D%DZrbrT  !  0>@  >D0  .& -C192C D%+P,Z7Dˆ6 pDLcD9+AZ+!h*b.>^ D.7b2JEEK >IM("   E'0a9 M'PB0A @DBj' RWZb% e9S5 ?))1u> 0945A*" 0..5)MF*)Coverage as the signal strength is varied* C 2 A:VPr@  #enC/ < "3!  ) %0Ǭ!5 DD!i!TZ_-b9EB/g9e-Ah:1 0B= 1"0..,5X*2 13*1%VaZ)fe e.e %g2D* ںں2Zm\ڌD! 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KoDoQAh -W* $ AA^R2Z 2 . 1"0 "8 [* m( *N J$=v$   H  "decimalb  | 4 *!]E0D* *JG-ZPK IVYIndex/Slide-3015748.iwaINĈG"  -*0舸шވ눸ӈ "31B/ Transitionnone?)?0X*2ވ:舸:ш::: yHӈGJK C)<) "! !   *܈ʈǬՈ  R  DD %%uh%@A-?0(8RTZ_`h1bhh*b EB/gBT e Bu6BB >D5 "00  ., -EdW*4*  2 :.Pr en , C%ٶ02!70A!C$@DBTZ_-b=S5 ?))1u> 0941"!..,=i*2LStatistical problem5-7e0ib"  ew ev8qsPv+C4C EDC RQZ\1Ab.11.C1@<@ae/g]j@TZ_ b@% SU V\Mh">2 0Y4Q"A.3.$,@*E!I>Model the data as a mixture of background and signal denesU`ZB"c ( ) . 4aEL ZMbf" Ae.we łgmy @ 2_}Lf#BCB bDf8f=}a QZ\A=_"K8JqXZ_KDB* Helve( Neue:AϥDk_?` bCJAoX \sim F \ : \ dF(x)=(1-\lambda) \cdot \color{blue}dB(x)\co  ack}\ +\ 261@red}dS_\theta(x).%.7<0a}CK{"DeQZ\!_b.1.C 1㻅"!ڑ _Cve jTZ_ :S T` "> 04".3.,1f*Sirtectionr*ZrNC%-2 C1 TD%2TZ_VS5 VU.%-"!.%)0941"!..$,p*)p;construct a confidence interval for j,strength 5]ZBm (} 7J 86  8F 3  *! jc՜ ~d )bեѥ.9 MAˡAYr! J" QZ\Ar٥Kť Z$Q$B*2-Light'?d.q.?8" p6\ (>NU 1D? D9OZ!OYZd!Ogjmp&.?.C ?cc. uDZ0TZ_ cc".3.,c_*2c$ This talka#Za6?rT%-2(DGs5A0e>\zTZ_VS\ }*<">AE 0\4\"!.. ," * l ͖aQX,5We focus onXding >esf.RSZ= 2o6.=2N=36= >==R= uC z=!j=QZ\AS= u==.R.C9 !" +D{֩TZ_!V>ieagv"a.3.$,* *uy0,Characteriseaw fixed fun  ofn  j &q6 *    & ' NR61>Rq!jQZ\Am.l.%6j>=~  1vJjCR8 *} zQZ\6K ZvjF\to(F) V X9@Bl..C z蜩za6z2zTZ_!fzz"z.3. ,*# *# v )QuantifyPuncertainty due to us0  hc6h. &NT36i>}TuN*C0}!jTQZ\AGTKe Zi \hat{F} . rry.p.E >s~   @@2C&jWZb!CC"0.:..*}FE'Isa$re any new4 damentVarticle?&: YYE - FC%"o!WZbjS"+ u>20&4"0...=I*&" J+.vFg=!6qԎa  D !oDoʉ!-W*;$ AA^{!D {! or{!"0 "M[* #]!.!J^=v$  HH  "decimalm#  # 4 *!]E0D* *J-ZPKVIndex/Slide-3080548.iwaGUR䂼K"  -*4ゼ 7d"31B/ Transitionnone?)@?0X*W2Z:}:::::x NOPQRS >7)("    @ *낼Ȭ2VV R  =C MDKD%!h%@A-?0(8RWZb`he" K*t8J ~CT}CZVbVT  !s  0>@  >D0  .& -C192C D%+oV+A Z+!h*b.>^ B.ZDl  XD  JCEC<I "a8aA   E%0肼Ǭa(M"$DD=" RTZ_%b9 EB/gBb  B.  B. N 5!*" 0. .3 -EdW*6* ` 2 !:.Pr en , Ce02!70A!@DB!TZ_-b=S5 ?))1u> 0Y41"!..,=w*2!Approach 2: assume a signal model*1;)ءö02Lu[Dk9C [oDTTZ_-byS5 2M-"> 0941"!..,u*2-ANL a do joint optimisauG? 8LpR\ñnT1DS! EDCYZd-gjmp&.b?.C ?IE I /g]^@RTZ_ :%.SUV\Q">2M0Y4Q"A.3.$,U*E!R>M~ the data as a mixture of background and de esU`ZB"l ( ) . 4aIZZ" kf"  e.  e g" 2k D* 2_Lh#BCB bDf8f=6 QZ\A=_"K8J*Z_KDB* HelveticaNeue: kh?` bCJAoX \sim F \ : \ dF(x)=(1-\lambda) \cdot \color{blue}dB(x)\co  ack}\ +\ 261@red}dS_\theta(x).%.Ce}{(D* _CeVzTZ_!>Su\m".e[  yq"a.3.,un* u8 ( polynomialy)^aC%" *p51TZ_VSb"!-"!..,=k9ӡ.1gaussia&UB?bXšˡߡ᡾2b? NbYZd-gjmp&.M?.7 ?0"6 bR6 4ޡڡաAESy *ZS. QZ\ _ .1.C 15⡾ݡ ubBAݹ5!hZ TZ_ "ab.3.$,* j 4Fit* , * Dstrength together  &  & aAB*  顾̡ d B+ 6D_f3> QZ\A3& KB d J &BT_{MLE}(F) = \arg\min_{ ,,dB \in \mathcal{B}} \ \text{KL}\left( dF\ ,\ (B] dB +:? "4 \right)%.%5.Cѡ܍2p7B\2i9 C&=jTZ_!F9 Sf0o~$ ">v0&9 49 ".3.,6f ? Problem&6 &~6 BCò桾-2 <C1 jD%;zTZ_VS5E-".%* 9/41"!..,**䳸%3 3*y ma| fit som)MZϡ-2גC7aST ZSb"!-%۪! %.. 3" * !aԡ5%if %dP is over-parametrisedJ _6  ^  Vj;  *! jc"O~d )bΡe!%, -LBˡA6! ZQZ\6K(Ze e,=0Q$B*2-La'?f.N.Et >"    ?*6 9_䂼!_NeWZb!_"0.:..*}F'Isa|re any new fundamental particle?&U ""ER FC% "DWZbjSJ u>2 .z Dz"0...=I*&eJF+.XF= 6qԎ!  WoDo`-W*$ AA^^*  rR"0 * [* x*:N;$z$  L  "decimalf"  " 4 *)]E(D*# *JG-ZGPKiV{I7)S("    D *ȬWW R  y `C [DMZD%!h%@A-?0(8RWZb`he"K*t8J ~CT}CZWbWT    0 @  >D0  .& -C192C D%+oV+!Z+!h*b.>^%l !  JEE[ >IM("   E'T  M'PB0A @DB' RWZb% e9S5 ?))u> 0945!/" 0..5)ML*3Signal detection requires estimating the background* M 2 A : `Pr en , CE < "_!  ) %0ɹ!5 gDdeD5TZ_-b9S5!A-"6 0941"0..,5~*20(We only know%(theorized sIregion*1B)ȹ-2FWD; !O6TZ_VSB"!-%ϩ F1"!.. ,5* %Фu8is! bump a%0ificant deviae from &u?5VZBa v   0Ǭ2  DD|TZ_M b}EB/gf VU 2n "A ..,}W*E35Z  YDx !D?TZ_-:Su L%*m"> 0y4q"!..,=p*i11. Esse:54) -FNa'TZ_ #τ.%с-%0941"!..,*\ F,92. Check if)ҡ(is far away est.Nb ) ⥽䥽b 饽>I5lO TZ_u~-ua.%)0941"!..,}f*2z>L u*iĺ%=DDaa^ TZ_VSaoet-".%) 91"!..,}*2/D, nce test [Algeri ’19], LRT [Cowan1] 5I`U [($50A7F3F7-44E8-45D4-BB38-A1127177E41F  (&$EFD4B01B-D53C-4FF8-BD49-ECE49000148A 'Bs E2ee BeC" WZbMONeS]OQ/u>2 0.O DQO"0.?..I*&dUJ+. F$=6Ԏ!   oDoaaJeO9 -W*+ $ AA^2+ s B2 1"0 " [* *NJ$=v$  H  "decimal1 zK 4 *!]E0D* *JG-ZPKиV2e/ Index/Slide-3093238.iwa >if_"  -*HѾѾܾ Kd"31B/ Transitionnone?)@?0X*2Z:: :::::::Ѿ:l xz{|}~ >7)("    @ *Ȭ2WW R  (B mdDfcD%!h%@A-?0(8RWZb`he" K*t8J ~CT}CZWbWT  !  0>@  >D0  .& -C192C D%+oV+AgZ+!h*b.>^m!l !  JEEs >IM("   E'a@  M'PB0A @DB' RWZb% e9S5 ?))u> 0945!/" 0..5)M>n*#Approach 3: assume a control region* = 2 A : PPr  CA < "|!  ) %0!5Lp\Dd D [oD?TZ_-b9S5 %*-"> 0941"0..,5*222. Extrapolate the background tosignaBL) enN#-F hCTZ_;ʨ-%>%ک0. D1"!..,=J/1. Fit%=withou n-FlTZ_.%i-%1 . D1"!..,=J,43. Check if)Lump is far away from u赲aa Z" f e.C N e g" 2D* 2Z4eD@A Cf6 QZ\A_" K8J* ZZKDB* HelveticaNeue:%Ak!?` bصFB,\lambda(F) = 1 - \frac{1-F(SR)}{1-B^*_F(SR)}..Cd r<ǬCD!kE r;  TZ_!kbCEB/g.C euR :B 0B6r DqrCer.3.3* W*2B8* ҾѾҾ=+m TZ_--"*.. ," * { " 070. Under our p|, outsid 2 a5UZB  3u. r aBBѾҾBfBD8hAqD\(BQZ\ABKBBfBJB@ A6dF(x)=(1-\K) \cdot dB(x) \quad \forall \not\in SR.=.a=ܾ11&_D!(iDͬCZ>QZ\6KBN {BdŲ\ = \arg\min_{d\tilde{B} !|\mathcal{B}} \ \text{KL}\left(dF\ , \N  a$} \right)%.%).Cptåtýt=D4,CfU.g,T,%>tS*ot".  2 6,"t.3. ,t* õtTProb. that  falls e X9ZXJo U6Tli    Fu; _ *!  a}Yf db íj%, 5BAy!VQZ\6KBdjX\sim F QB*2 -LaI <ǟn?% >->=52 R_.3.Ci.ǩDCIXaUCMYTZ_!RFSb"!mz  j"a.3. ,ZܽJUڊ;6!6jc"Ymj~YɼʼqN'DB. #]A!jQZ\AA& KBi iB_F^*R&qA6i.J.EjppBpC "Dj0WZb!eZpS}qu>2 0. D5a" 0.:.5*^ I*{&pJ+.yF=6!Ԏ!!  oDoOږ-W*$ AA^2 2A.:"0 " [*  *l!N$=v$  yH  "decimalT  n 4 *!]E0D* *JG-ZPKV0Index/Slide-3095408.iwaJC"  -*,ӾӾ "31B/ Transitionnone?)?0X*2Z:u:x:{:~:8   GHIJK >p7)("    @ *Ȭ2VV R  (B mdDfcD%!ih%@A-?0(8RWZb`he" K*t8J ~CT}CZVbVT  !b  0>@  >D0  .& -C192C D%+oV+!Z+!h*b.>^m!l !  JEE; >IM("   E'T  M'PB0A @DB' RWZb% e9S5 ?))u> 0945!/v"0...)M@*#Approach 3: assume a control region* = 2 A:PPrP  en C < " !  ) %8Ǭ= DD!i% r !TZ_-b9EB/g9Y.AfQ 1 0B= 1"0..,5W*28*1%!NͶ8ӾӾӾ=Lp\D+m C [oD?TZ_-b=Su ;ʨm}"> 0y41Ӿ!!.. ,5* % q070. Under ouri`ption, outside the signalm (SR)5UZBa 3u6jaZif e."e eg2D* ӾӾ2fD8hAqD\(f6QZ\A_" K8J uZfKDB* HelveticaNeue:Vka?` b@ A6dF(x)=(1-\lambda) \cdot dB(x) \quad \forall \not\in SR.=.E 66NB6C!z "D6WZb!z.S} B "> 0*B Dq"0.:..SI*5u]J]+.LFE=B 6Ԏ!   oDo-W4$ AA^ 2 B5r"0 [*  *NJ$=v$  H  "decimal  n   4 *!]E0D* *JG-ZPKV>'eIndex/Slide-3227773.iwaHa^W"  -*@ɪΪЁɁ CFd"31B/ Transitionnone?)@?0X*v$2:: :::::xɁ \r^_`abc !(   )V] #, All at Onc"<  XN@- In apple:appear-"1"4X(w  oO )p 6! ]?(0: -BR>%t7 a("    @ *ȁȬ2rr R  *@B 0Dh؁D%Ah%@A-?0(8RWZb`he" K*t8J D%DZrbrT  !  0>@  >D!  .& -C192C D%+ @, 3D#rD9+aZ+!h*b.>^ XDl q2JEe* >IM("   E'a~ց& M'PB? @DBj' RWZb% e9S5 ?))-"> 0945A*" 0..5)M$p*AlBackground model selecK* {4 2 A:GPr en , C < "!  ) %0݁!5LD4C 5FD/gTZ_-b9S5!."-  91"0..,=w*2\!Estimate without signal) ling5;; 4im2P*ʁq]C]![CT:ǖUZ`-cfi.J8.a 8ZՓMfy e."e Egղ6@ Ձ2h SC cK>6QZ\_"K8JݏZhK(B* HelveticaNeue:ǟn?% >->=5+` b!B+\frac{E[\lambda(\hat{F})]- }{ .s.C)&-j&0ҁ&QYal(P:7B >!v4C =j. TZ_!jb4*..S >k uq"0\.u.,} I*6uFae%]-vUQk8C! . C/A%!`zQZ\!YKBN1,? M=0.01..C ]܊  ˽dDTZ_!K".3.,}k*ӭ2*$complexityu'/'ra }2 2-Z-%D=r Cf>-QZ\!y2KB2Z-DR-Ub -R FB,m2$(F) = 1 - >81-F(SR)}{1-B^*_ ..C.R8Ǭ}RD~aRdBۄhfk>.T%kb*T SbT  > 2T nT "aR.3.,}RX*2R12u??70"5 bR5 <–D&aTD@Yi*j 1 QZ\a_b. 1.G 1@~Rt a4 ( DD|W ZaZl orux{~4.M.> M$2WWV9QC!~jWZb&K*tD~CT}CZWbWv0 @  J .1.&oQ 6+!Z+>5bomd2fc2JCI` 0a" pa̲) %Bm^RTE" . S " \.u.,G N Ciϲv-v 9v `"CvTZ_av%v"!v\.du.,vv>9v 0A"ՄC̡g*ATZ_-vb*gS}*.P CcBB)y* . D"!..,6ZcFCᲪ)20CN֪TZ_"!.., 2 : ( X3D* jbaTTZ_-:S}T mT".  .T DqT" ..,"*2 (6Choose the Dsmallest extrapola error* P r3C` )2rCX ygDq=j5 !TZ_VSb"!-  j"!..,=*2 don8 [Barreca’11]" 8 enA  2 ($2F30A774-3FA6-4141-AFED-EA8BC6B436B1 BEBCN {"bjAWZbM.bS]1e> 0. D"0. ..6&5J+.Fd= 6(V!  oDoae0^r W*$ AA^*- 2.:"0  [* qk*=NTJ$=v$  VH  "decimal"  " 4 *!]E8D**JG-ZPK渘V_!Index/Slide-3228122.iwaIFڃ?"  -*(ك +d"31B/ Transitionnone?)@?0X*c2Z: :t:w:z r, @ABC Eo > T"  "    @ *ńɄ   R  B0A @DB%!ch%@A-?0(8RWZb`h1eh*b.STT  ?)) 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