Perfect Forward Simulation via Simulated Tempering ... - probability.ca

Download PDF
1 downloads 0 Views 339KB Size Report
Comment
sampler (Neal 2000; Roberts and Rosenthal 2002a); simulated tempering .... to " trans-dimensional " (or " reversible jump " ) MCMC algorithms (Norman and ...
Perfect Forward Simulation via Simulated Tempering Stephen P. Brooks University of Cambridge, U.K.

Yanan Fan University of New South Wales, Australia

Jeffrey S. Rosenthal University of Toronto, Canada

Summary. Several authors discuss how the simulated tempering scheme provides a very simple mechanism for introducing regenerations within a Markov chain. In this paper we explain how regenerative simulated tempering schemes provide a very natural mechanism for perfect simulation. We use this to provide a perfect simulation algorithm, which uses a single-sweep forward-simulation without the need for recursively searching through negative times. We demonstrate this algorithm in the context of several examples. Keywords: Small sets; Regeneration; Coupling from the past; Reversible jump MCMC; Band-return data; Autoregressive time series

 

   !#"$% '&)($ * $,+.- /021 134 "'5 76 $+.- 89:/;7< =  >.?@"A 6B&)CD"'$FEG"AH@0 $IJ $%6K !7   1. Introduction

 MNOP"Q46 Q7 SR"T"'VUWV+"A6IXY64"Z[$% > 7>\ ]"'$"A W ^ _"'$,P7 XRV $ "5W+E) (( Y V, 7 XVR "$JQ7" L (7

WXEG"'I`+ "' $, badcM$,Ke4 gf'11 8 h5a~^ $, 7 $,k 76_C) $Y F)fA11]/4| €Hƒ‚S‚'„W…†W†ƒ‡,ˆ'„b‰Wˆ'„F„W…†GŠˆ'‹K‚V…‹4‰F…HŒmŽW%Sg‘’W“V“A”•%–nŽF—5ŽW˜t•jŽW˜š™H—›2œ—'4“’F—ŽW“'’ŸžV– M¡ ˜šS,’W•Ÿ˜¢Ž[ž*“£ƒ¤D—¥g ’W˜t¦§AA–¨p˜š›š4,’Ÿ£Y“’W™% © “A—A¦–4¤D—¥g ’W˜t¦§AA–K¤D‘DªT«¨p‘ –] ¬ ­b¬ E-mail: [email protected]

2

Brooks, Fan and Rosenthal

'" +"'$! 76 {JQ P(7 F  f'114| y   76p4 7 Vf'11 1 h53 76N,$, 7 ,64P"'7 F  ƒLML P"5,Q46 la $"'" fA114|4cM$Ke4  )/; ;}]hH< Ÿi T,QY (7 ("'$'3SIM"ƒ"5r \ "),Q" ((  ' V, 7 kE$"'>"7"$% V R") mY V"A6g"'\(#"$, > ,%Q"P"A   3  mK"AH, \/43SK+b],$46477>XQ"J5 75"'(4, #"Q 76!  !Y V,"'6o "'  > 76b,Q" ,5 ( >@EG$ `,Q" (  F ajM0D{XC@|SCƒ$,((o 76o9x  lf'11 - h#(#"$EG"'5D mY V m %Q"'P"3V  I "'K    ,$4645 >XQ7"  7 E7 *   "5% 3]5 ( >T 76\$"'>"7"$% V   %Q7"P" 6T64"' ,5$,#")WIM "(7 $, "D$"'>"'"$% V,R "D ,%Q"'\"A ! m  7 )' P#"5 ( "'6B3 Q$,7>QP T(7 $Y57 $)$"EG$,!Y V, mE,Q"  !Y V,"'6l"P(#"$,7>.  >$,Q_ "5ZŸ FIM""'(.EG $IJ $%6KZ[ \"T mY V _  > $Q7v"'"$,YP("'$FEG"AHm mY V  ,%Q"'\"*  "'6 (# ,Q"vM0D{XC ,%Q"P"

76p7  >,Q"v$"'>"7"$% V R"*(7$ ("'$F,"A !E  mY V"A6p,"P("'$ >N %Q"'P" Y $,>"'$ 76!Y $,>"'$ RV  "A D#E "o $Q_3 76! P"IXQ VDP($,VR"A , D5 P(4,  7  "u.5 "7+< Ÿi lK"AH, $bI "X"rK"6*  7En,Q"' "@Y64"'

DT,Q"X,$, 7 WZŸ64 P"7  7 ]"rK K+!"5r \  >T,Q" F$,>   e4 D"WI "'"mQ"X m "A6o,"P("'$ >T 6!$,"R "$% F  "kUWP(*LML %Q7"P"' '< 9N"J  W,$, "MQ"A F"JY64"'

  Q7"5 ]"5rKoEX "R "$% )"r4 P("A 32 75 764 > ^  \(7"."r P( ". 4"'H,  -< f3D v$% 7 FZŸ64P"'7 F  

4Z[$,"> $"A   R"o"r4 P("P  K"'5   43 76N $,"' ƒ6 V% _ 7 +4  g p4"'H, N-7< /"'"$% B64Y 7 , F vE2Q"T P(  ' V  X 76>"'"$% B5 ]"5rK@E Q7"' "T 64"A  '
7\, e"' (Y ""$% ' IXQ Q"v5P( "5,* " #ZŸ  >"'$, 73D,/ +ƒhH ""'$, ƒ$"EG"$,"7"lP"'

F7$"A ! $,".(# , F  "3D4\EG$P5Y $,W+LIM"l$,"' F$, 5o y "'"A F> ""bP"'

F7$"bD  _37$]  "'6lIXt,Q57 ,7>\P"A  $"g  mhM> R "]+ ƒCa "O7h 1PQ.  F  a ƒh53IXQ"'$" 

PRTSU .Q 4@,Q7 VgI "T +vQ7 SR "o4Zj$,*   "'6_64"'7 t,"A 347"!Q7 Eƒ$,*   "'6 64"'7 t,"A @ $,"g SRV Y  "T 76v $,"b7 "'6v #VR"3K,Q"D  PR f < ((  ' V  k ocM S+ "'   4EG"'$"'75"ƒE "'o7 " X F(#"'t#ƒP"5,Q46gEG$ Q"'  > M,Q")% $,>"5264 F$, 4  3 eKVIX  Q"'  >7#( 3IXQ"$,"PI "\, e " F a ƒ) h ^ ƒa  h O_` EG) $ :6 .p($,(# , ($ 7 7 t,"A4 e  a d"Odf hH< a "+ "$_ 76x{JQ7P(7  aWfA11 h bW ca d"%A h.1 >> "' F,Q7 VoIX= Q gh! ,f "i$'$&$'" ")(]3nI "*P >Q]o F")  e  a d"Oj fAk h le  ca m"O nOfAk h  o " EG "'"$% M %Q"'P"' \ $," Y L5 P\ k^($,(# ,  6"7 t,"As e o Ca "t f h IXQ_$,"' ("AHX y "#"' >"bP"'

F$,"
 ,%Q"P"*IM$,eK b

gEG VI@ ' .$% 7  : ,%Q"P" aj Q ` 6 $"'"K#"$,>Of'1 1 hm,  4, 1     lEG  VI@ < 9N"N"'> OK+ > ""'$,  p >  fTb o Ca Q"iA¢hH< 9N"_Q"'

"(4. 6 "53    T f IXt,Q (7$ 7   tW + o Ca "t f O|   * h  P  #f "tJ o Ca "t f "O  h IXQ"$," J o a "` f t| 7 h  F F  7a aC ƒfYhBBh e e o aCa  "`f"`ƒh h $ a~/ h '

o

f

,Q"$,IXY F" 34I "T "5@  8,< 9N"o F7\* $Y F"g,Q"o F !7 "'6l"'P("'$ >.  > $Q7  MEG  VI@ 
 f @#/%.3YX!B .ZE.;p #   WT > * - #/.10DP"*5QSRTK   Y (# ,  "3 Q7"xIXQ"'"R " $    Ac3JI "^'  (n6 V,"   IXQO   764"'("'764"' l6$, SIwEG$   a NQ7  ,Q"l7"I RV  "3   V32Y o 764"'("'764"' \@E   3) 76N,Q".EG$,".(7 Q EM,Q"%Q  p !76"(#"764"']mEJ,Q" (  F' ""'$,    P"' '3 6pQ7"v "> \"'], mEXQ7"v , P( "( VQ "WI "'" $,"> ""'$,  ,P"A *a dbe;fSgFhg $,"P F4%Q7

W, '  +_ 764"'("'764"]la~+Ke]Y 6   f'1 1  hH
5"' '3]E"'t,Q"$JY64"], ' n$J64k i#"'$ >P64P"'7 F  _64Z P"'7 F  PQ"o "7>Q_E

 Q7" 7$4Zj ^ad5 KR"$,>"'75"Shk(#"$, K6k ""'$, "m%Q  7 F, $F,>T 477t," +mEd $  PQ"(7  Fƒ$A3P $"@(7$"A5Y F"'+ 3m64"5,"$,\ "@IXQ7"$," F%QP%Q  7  dYe XQ7 SR "X"76"'6 ( X \ " '"$,7< C $ ((^ 76^9xY F  aWfA11 - hX64"A $ #"oQ7 " ?dbe  "f d  bg MajM02{XCMh > $,t,Q EG$b 6 (4 > LMNO mY V _ m,Q7 V@ P64$% SIOEG$, ,Q"b, $> "5X64Y W,$ 4, _' v"T> 7 $% ],""'6 VJ P" :;3 76 MY M(#.,QY ,Q7 VJIM"g Q7  #7  " $J  >$,Q_Q P"\ W% V"\ V \ " RKn 76IX 2Q7"$,"5EG$,"

 #"*  F, "  c g,P " ) ; $Q (7$45"'"'6 J

MEG VI@ badC $ ((_ 769xY  fA11-]hH
  KL"MON d; "f EG $!b/ 6U  U nOfSh@  (Y 5"\E nOf\ T

H  +P$," "u.5 "]@v>"'"$% ƒa~Cƒ$,((v 76v9x  Nf'1 1-]hH< ' "vKRK 7 m($, "zIXQ P( "P"],>LQ7"^M0D{XC  >$,QzY !,Q"_""A6p,L$,`ad("'$Q (7 m$"Z (#"' "A64+hg L7"PK!#"$bEM%Q7  7 EG$  "R "$,+_7" RV  "\ F, $F,7>l(#  A< ' "\IJ S+_ $ 76_,QY (7$ "'wY X,.7 "TQ"m 6"' PE)P7 Y5W+l,. PtQ"m]7!#"$Eƒ%Q7  7 J,Q7 V7""'6v,.#"m m "A6B< 07$*"r4 P(" 3MC $ (( 76 9xY  aWfA11-]h!7 F$,75P VIM"$* 76 (7("'$l%Q7  7 PIXQY%Qx W% $. VPQ"_WIM "rK$,"P"' !E@ 77t,"l(7 $F,  +]Z[ $,64"'$"A6L F, "l (7

5"Q7"$o,Q7  K+ Q"'$.%Q7  k|  \ Y $,+ 3ƒQ7"((#"$l%Q7 x IJ S+4 m  "' Pp VI "'$\Q7  K+ ,Q"$.%Q7  < $"'"

76$%644%QpaWfA118]hH"'"$%  ,%Q"'\"TE)>"7"$, T ((  '  W+.,l *IXY64"o$% > "TE)$,"'  Y W,  (7$ "'* MW+](7 ' BEDLML: ((7Y  7 JQ7

J+"M,*"o6"R"' ("A6B< 2.3. Small sets, Coupling, and Regeneration

?"$,"*I ".6 ,57 , o *   F", bEG $!^ $e VR^%Q7  7 'wY g    kaG $'3k&a %YF,"('/"*)hŸZj *  YhME2Q"'$"b"5r4Y W% X \($ 7 7 tW+l\"A  $,+ " )BOa A hX Ca  "  h53

\(# t,R "b]"'>"',$ %YF 37 76 '#. ;P F%Ql,Q7 V@IM"bQ7 SR"gQ7"   /dbf $g   d &Sd    d 1

[< "< 3tEDIM"b IJ S+4 MQ7 SR"

0/ a" /!h 15 '0)ka./!h" +Ma.



0/ 0/ 

"

746

" /  

 1ƒh152'3)Ba /!h

ad}]h

EG $*  h"3 J,P"_ 764Y5"' H3M 76  J\"A  $%  "v 7 "5% )/sEQ7"_ F, "_ F( 5" "5 !, 5$,$,"]@ W% V"o 76_ F P( +oUWP(_P,Q"!64Y F$,  )BaBA¢hH3  > $ >^, \57$$,"]\ F, " < K%Q *UWP(  \$,"5EG"$,$,"'6NN  \ $"'>"7"$% V  v  75"lQ"_%Q7   "'> 7

>  # !EG$  Q7"o64 F$, 4  )BOa A h5< K* ) "5, g $,"m  l F"EGDEG$b5 ( >v!7t,( "P5(7"A EƒQ7"\%Q7  3#IXQ7 %QN   Q7 ((#" N"v Q7"v *  J "5 " VP P" H3 Q"' IM"'  7 F$,75m,Q"5(7"A @UW ] + ,Q7 VPIXt,Q ($,  W + 'A3 V\ P" j %pQ"'+ $," X  ,Q"  P""r4

H* W% V,"NadIXQ %Q Y m, "Eg%Q7 " $% 64P +

5 $,6>P,  )BOa A hhH 464 tW+ 3 IXQY%Q64K"A @  IM S+4 XQ7Y6B3Q7VI "'R"$Xg6]"A  P"5,P"A Q  6vEG $$,"' Y W, o\464"' mad ""m"< >7< =  "]Q7  fA11}]hH ""'$,  \ P" "F< 20  7   +3AI "ƒ64"7"),Q"?M dYe;fEKQ"ƒ%Q7 g,@"D,Q" "'q]"'75")E]$% 76:RV "A ƒa   "    q"i$%$i$ "    hH< 9N" 7" Q7 LK+ 5 7 W,$H  3TQ"A F"  $, L $,"  764"(#"64"]p 76 Y64"], '  + 64Y W,$ 4,"'6 EG $ZM: ] ,f ",d/ "d} "%$i$%$š3JQx  "$,* *EgQ7"^"'>,Q EbQ"^7$. 76 ,Q"^64Y W,$ 4, xEgQ"^RV "A PQ7"* F"'R "' '< adijE2IM"T W% $XIXt, Q Fx F P( "N "P* a % e "`EG$  y "P* P-mE =  "]Q7 )f'1 1 ]hH< '



.3&C&C95("*,+3*e 1d g

bf `d f 1g d 1

 

1 #"

#dYfgd  /f d $  $ %0    

'  '  ' o 

f

         !  1

d

g  6g # & k g  $g! g 

a " o h 5

" 1' 

74 

a"iA¢h5 ' o B) aBA¢h

74:

g e;f  k) aBA¢h d  

   

"  " o "

g e&$ g"  $ g g 

Yf

fSg+%_Fs %  j1% o'

         

KRK 7 F +3 tEX  F77 F"o *a&%YFZ"('/"0)hjZŸ F* [3k,Q" o o   a %_F,"*' f "*)7hjZŸ F*  )EG $o K+ ' f76 'A< {JQ7 P"'  ƒ,Q7 VJtEI "Q7 SR" Te% f g       E 'A3]Q"'lIM"' l F B (( +P *  # "5M 6"'  ga ,Q7>Q. "' , (*),+.-0/13254 '

Perfect Forward Simulation

9

"u*"'] +4h5 $Q aG"A5"' , , $, +p"'P("'$ >]h53XIXQY%Qx lEG$,\ + "$,>464YpaG[< " $,Q IXQY%Q_KR  R"' J$,7>\,Q"!  > $Q7 EG$@ P> "P"$, '  +]Z 64Y F$,"'6vKm"'$@EDt,"$% V, 7 \ \M0D{XC > $,t,Q 7 F >!Q"b%Q7  D! 64"7"A6v

J> $,t,Q /4< }< y "5 "



1e% $ f "

d #  f    d 1g ' / f d Ye&  g

 ( 3KIXt,Q(  >m 76*$,">"'"$% V, 7

 P  ! 5 ;\& ! (T$,"> ""'$, "' : V:,P" n

%YF ($

J{ Q"'LK+^7 F$,7H, k3 "    M\Pa 'HhH3k  75"P$,">"'"$% V, 7 @4'5$bIXQ"'N],"(p/c Y b7 F"A6 76^,QY 4$% BIXt,Qo($,7  W+ ' 76"(#"764"'] +b  "'

%Qbt,"$% V  < 0$Q7"$,\ $" 3 " Y 764   +76"(#"764"'] RE ! !( F#" 0/    3k5 764 7    0/  767( ^Q7"$,"m#" >v.EG$Q7"$b$"'>"'"$% V, 7 JEG$,  P" n,%_F"6jOf,]\ \"o;7  >  "<   764 LaFfAhƒIX nE "'.Q  6B34 5" )kBa A¢h Y E "'l5 7 F$,7H,"'6*"rK(7Y5 +P!Ed 5  t% V"Q"gR"'$' V  EDQ"oP $,Y   5 764 k< a~?@"'$"mQ"* F, "* Vg P "  Pa   "O  h)  H *3 $, Q"'$,Q7 UW7 F  6_"7"$%  + $,$"A F(#6 , Q" Q p64Y W,$ 4, k3JIXQ %Q PrK"A *R"'$+ $, (Y64+ \   d a~ F"'""577t, }< fAh)$  c #"  >   *ad " " "5t, }< } h5\$,"' tA
7h 5 '  " 73:Dr"  D -$

!1   a"t7hQ" /6051' 3ƒ6 a./!h #dbf   7a"t7h 1H@     /   Hg/#+ '   1H@

  .&  $g   d '   

 

f

      #   &     

$g % Z j_f#"(' /"(6 h   @a

  "$g 76@Y%Q  \aWfA11 1 hJ"A  "]Y   +v!7"!QY $"A FIXQN@ > $Q /< }P,l 4,  l$"'>"'4Z  "'$,  R"  !Y V,"'6 ,"P("'$ >N  >$,QzIXQY%QpQ7"+ Q7"  F"_  \ ^7

FY !EG$mQ"'$\("'$FEG"AH* F !7   ,%Q"P" d"@  - c h8 ?  a a./mh" 7 / $>r"6D "  D -"  !    % d #     # e d #v > $,t,Q F !Y V,"' b v^ $,eVR_%Q7  ^Q7 SRK >_ W% V  $,+v64Y F$, p?!a"t7hH a"t c h * - #/.10 3(  *:QSR#"%'. /!87D!B #`9!8B`@/#/%.37.17'.3/9;#". a   "O  5h    aC  "O  h " A  HB !87 9030103"!$9;#". B !8&CH'`9;#".3G:#".3& Ql  (4 *  h),!7 " K+ e% !f  g       E 'g 6B3#

@IM"! Q7   "" 3EG$($% H, ' ($,(] F"A JQ"mRV  "oE '  a %hJ($,VR]Y64"A @ "i#"AH R"m((#"$ 76 (*),+.-0/1 4

Perfect Forward Simulation

13

 pQ"RV  "' m7 "'6p pQ7"LML  >$,Q* o64"A $ #"'6p pQ7""rK\ "'H, k^RV $, 7 TKP"$,Y  "A%QYq 7"' T,^"' F * V,"'A3($,VRK 6"'6NQ V! ".Y ,5 F"'$RV V,R " \,l" F$,"! ^764"'$"A W,* "  IM"M' ! , P( " 64 $,"'H,+gEG$, ,Q" EG7AUW ]2(7$ $ a ƒh5v ]+L64"'P(] F LEMQ"P% $,>"b64Y F$,  ],WIM5 \(#7"], g7"\EMIXQ7 %Q  L#" * rK P "'6v   + , '  + 76lQ"TQ"'$@  P("A6.EG$, 64 $"AH +< {JQ"N$,"' t_* S+ "LEG$Q"'$"r],"76"'6x, ,Q"   "IXQ7"$,"IM"$,"'q] $"L]"'$P"A64 ""P(#"$% V,$"A #"5WIM""' Q"LQ_ 76xY6O64Y W,$ 4, 7 '< 0$"5r P( "3I "^P >Q].% e1 " F 1 ad67 V,  ƒh O_` 2a )h.

6 ,57 , "'6   K"'5  /oo %Q7"'R"X(#"$EG"'HM F !Y V, k $Q }7"'"$% V,"'6N 764"(#"64"] + EJQ7"57$$,"]! F, V,"Ah5  f ZVca   "  hH ""'$,  >2K    K ;d"'fj3J 76 Q"' E K   U  acQ" b%|tWhmI "_ '5"'(4m,Q"_($,(# , bad 6 "52    f hH3 Q"'$IXY "IM"$,"WUW"AHPt^a~ 76 F"   -Fh5< $ %7\"']'3K~< "*  >$,Q Y X  MEG  VI@ '
_,Q"$,$"' ! F, "4^>  ]+ > ""'$,  > "    EG$  P"o64"'7 FW+ e o a " hX 6 F"F >3 fXr j " \(n6 V," E "'+"'$o 76p{JQ P(7 F  aFf'1 1 hgIXQ $,$"A F(#647>v

5"'(4, 75"*$% V,> R "NOaFfAh5PIXt,Q($,(# , Y EG$  b o Ba A¢hJ  M7 7 [l,QY 3 Q Vob"r4(Y5 +$"A V"A b  ~< [< 6B< Zj e" l"'R"]A3B IXQY%E Q   !Y b(n6 V,"'6  764"'("'764"' ,+E   3KQK7 @  VIX7>m7 X* ((7+lQ"T$,"' , JEQ"T(7$"'R]  X F"AH  < {  6xIXQ 5 ( >3@IM"64  764"'"'6:("'$FEG $ QY _64 $"AHv , P(  >pEG$, ?@ba Ba A¢hH3  "'

WlIXQ7"  K  UV c 37IXQ"$," c   ;7 /;4 E a Q#;  a d" #t| ƒh " aWfA; h +



 76=]  c  ! l&  acm" 4h1 c ( < ƒ r4(Y5 +l ]$,46475 >P,Q @\464' V _ ].@ > $Q7 7 < f37I "T 4,  ,Q"bEG  VIX>P  > $Q7v
 acQ " bW h@.3B.D`.3p #   T >  * #/.10D^"*5QSRTK  o U5 o   " o a  " "  hH|t   @`.3= #    > o aC  " "  h @.3B .ZE.;= #    >   * #/.10 "*  .;# > jOfE@97'GV".;#"H'"72#"5B#".30D('*

- #/.10 '* -

-









-







-















Ÿi T,QY  > $,t,Q_3,Q" "'R"]! ZVa  "   h8 c 76 K   UV c ( $,$"A F(#767 nX,Q" "'R"'  Q7 '3V$,"> $,6"A  E,Q"!$$,"]@ F, V,"34,Q"!%Q7 _IX UWP(_a  "O c hX VX,P" IjOf< 4P  $ +3E ] c  #; + ]  c  ! l&   ca d"Kh   c 7 2(]3nQ"'   B%Q7  IX #UWP(L a  `"t c h ,P" j`f\tE b ] c 76ZK   U c < {JQ7 J"'R"]@4$% IXQ_($,  W+ ' f  b  ,a ] c Oh c $ aWf fAh ijEn,Q "R "] 64K"' ƒ 4'5$A3,Q".IM"J7$% W (#67 V"@Q""P(#"$% V,$"X F >! T F, 76 $%6m"'\(#"$, >o(n6 V,"

Perfect Forward Simulation

17

7

F"A6^(#   76 K   3k 76,Q"N(n6 V,"!,Q"P W% V"\7 7>v LLML  (#67 V"\7

F"A6^(# "  6DK  o 3 UW7 F!  b p@ > $Q 7< f $,t,Q VR "@Y 7M .( P n" $,t,Q_3 ,Q"^($% 5Y 64u* "' . x \(7"'\"'] > - #".30 P VR "EG $*Q7"( P   > $Q7 $," E "N 7 F7$P7 %  " $,t,Q ' .#"$,"(7

5"'6PK+*]"'( : }4"'VI 76+"XIX  5 ] ]7",*($,VR]Y64"b64$% SI@ EG$,wQ"o$,$"AHX W% V, 7 $,+.64Y W,$ 4, k
"'"$% V,"oEG$, Q7"\$,"' Y647 e"'$"'  Oa A&"%A h5 "'6 76 Y P] W"A   +_ %Q "R "'6_K+$,"(7

5 > K   K+= c j`aWf n c h K   IXQ"'"R "$   D] c P("'$FEG"AH m  _ %Q7"P"
T c *  %'.37VE.;# > fE* - #/.10DP"*5QSR  " @B#'0T97G2".;#"H'"7`a  "O  h* #"%'.3_X!B. /7"#"!87'H.'* - #/.10\[*  /9EX K   " K  o  K*ad;d"'fAh @ b  b  aBA¢h@97'G "   b o aBA¢h* - #/.10\^"*VQ R b=] c E.;#K   > c j aWf nT c h K   97G /7"#"!87'H.'*+#"%'.3_X!B.@".;#"H'"7#"VB#/.10=[* - #/.10 "*5QSRTK   U5  Q"  a " bFh5|t @E.;#p  >  acQ" bWh@.3B.D`.3#p  >T * 0 *5QSRTK  o U5 o  " o a " " W hH|t  @`.3= #   > o aC " " W h @.3B .ZE.;#=  > * - #/.1 0 "*  .;# > jOfE@97'GV".;#"H'"72#"5B#".30DP'* - #/.1 























,"PQ7 oQ"*RV $Y  "' k.@ "i$'$&$'"O. $Q7 7< }lY T P( + EG$,IM $,64Z~,P"!R "$% F  ad

  {JQ7"$," /4< fAh2E M0D{XCpIXQ. ! F !Y V,"'6\,"P("'$ >!%Q  k"TE  6j in"$% J"WI "'"v"'P("'$, $,"' '< 5. Differing Supports, and Trans-dimensional MCMC

4((# "oQ VEG $"A %Q_,"P("'$, $," E l37,Q"$,"!Y  64Y W,$ 4,  ?@aOA h64"7"'6^ P"m F, "! (7

5"  7 $,t,Q* ad$,P  76 02 7VR:f'1 -173 Cƒ$,"' F :fA1 %!%43 $,""' fA11 h!EG$.7 F$,7H,7>  $,eVR %Q7  ,$, 7 t, 7 b"WI "'" W% V,"' TEJ64k in"$,>64 P"7  7 ' ,%Q"P"*Q"'$" < ?VI "'R"$A3 IM"."*,Q7 V\ F PY $oY64"A  oIM7 6p (( +"'q]7  +NI "' ,Q"l ,%Q"P"' !IXQ"$, "  = 6 `IM"$," (n6 "'6_ "(7 $, "'+*E,Q7 V@IM"$,"T ((7$ ($Y V,"
7\"']T

T#"5EG$,"<  R"N,Q7 V!IM". $," $$,"],+T F, V,"a   "t  hnI "D,Q"T(7$ (] F") X"'ILRV "DEG $ TK+> ""'$,  >   EG$,:Q" 6 F$,74  b  aBA¢h

76p F"F,>  f  ZVa   "   h5 *32Q7"$,"*"5r4 F, " f 8  7%Q Q7   o a  o Ca " " O|  f "O7Qh " " f O| m"O f h 8J ""'$, " "   b o Ba A O| df hX 6 F" fX  o a Q" " ,t| d"Odf h5  37 76 a   ca Q" bWh  o a  " " HO|  "   ca  "bFhFh $  ; a  "O  hQ"'a "  "   hX   a "  "   h ' $X Y k > $,t,Q_34>"7"$%   7>\> $,t,Q 7< fb M,QK7 @  MEG  VI@ '
  97'G - #".30)P"* QSR K  U M a "  "  |t  "t  h@ E.;#   >  f 97'K   W>   * 









20

Brooks, Fan and Rosenthal

- #/.10\[*



.;# >

jOfE@97'GV".;#"H'"72#"5B#".30D('*

5.2. The Perfect Simulation Scheme

EDQ7"o$"A F7t% X($,"' "]"A6 _Q7"o($,"RK 7 X "'H, 7 X"r],"76_IXQ74%Q > "bPQ7"T$% 7 FZŸ64P"'7 F   ]"rK'< ƒR ".,Q7>Ql7V8 I  74 6  $"g#" >m7(#6 "A6 m, 7" F +^aG$% V,Q"$ Q7 v "'q]"']Y  +hH3],Q" 6"57 _E[< [< 6k< Zj e" 37 6.,Q"TP $Y , V vRV  "' JE)  $ Y $,+.}\,Q$ >QvIXQ 4%Q7 7>"< {KR"'$F> $,t,Q 4< fo],l .(#"$EG"'5g m  ^ > $,t,Q_3IM"! >]  $"Aq]$,"! *RV "mE 'A _K+l64"7 > ]  c 7= 6 ] c

JvQ7"T($"'RK 7 X F"AH, v 76 "5 >

c   ;   /;  a 4E; a  #; Ma " " #|t"O7h $ +    

{JQ7"_I "g% e"

'

 b  a,] c ;h c $

aWfA} h aWf ] h

 ,"!,Q7 VA3B aWfA} hH3BIM"o% e "!,Q"m 47m UW7 FTVR"'$@Q7 "O 76  ,Q"\$,$"A F(#647> F, V,"\ (7

5"' '3 ,QK7 @(#,"]Y  +  VIX7>PEG$Y $,>"'$,']+_64"7 >.,Q"m F, "! (7 "' J, SR Y6 7 6# P( ], EG$Q"m  " IXQ7"$, " L   L37B  f @ .3B. `.;#   :>   97'G   W>   * 0 *  .;# > jOfE@97'GV".;#"H'"72#"5B#".30DP'* - #/.1 









5.3. Specialisation to Bayesian Models

 #"5EG $" 3SQ"M \(7"'\"'], V, oE7Q" $% 7 FZj6P" F 7 (#"$EG"'5D mY V m %Q"'P"ƒEG$2cJ S+"' Y TP464"Y ' v"o  \(77"'6vRKY !,Q"bEG  VIX>P> ""'$, Y   .EDC $ (] F v}< }
.Q]mQ"' "5.RDg. FEzf79/_EG$P"5r P( " 76  F"v@ >$,Q < }^ 4,  64"(#"76"] N 6$, SI@ bEG$,sQ"vcM"5, 6 F$,74 pEJ ]"'$"A WA "P"$,  6 F$,74 _IXQ_(7 $% P""$b ,Q7 V@IM"g, e k " .  O;m$ 1 ; / Q / IXQ` , K!#"$Eo"5r4(#"'5"'6 "'$,  7 l"'q]7 @ /;< /;3M~< " "5Q7"$ƒIXQP,Q"5 $$,"' (#764 >  4(XEG$  P W% 767 $%6lcM"5% P$, 764 RV $Y  "b>"'"$% V,$A< 9N"P(7 $,".,Q"l(#"$EG$,* 7"lE@$P  > $Q IXQ ,Q"v("'$FEG"AH* F !Y V,   > $Q7 E $"'"

76^$,6K%QNaWfA118]hH3]IXQ"$,"Q"'$@64Y F$, .EG $ ":Q7

M mP"' lED $,$ 6 /!t,"$% V, 7 J 76B3K#"'' 7 " ,Q"+P,  *,Q RV  "@K+P$"'("A V,"'6* m  k3]$% VQ7"$ƒ,Q7 64$,"'5 +\EG$  ,Q"> "P"$, @

ƒIM"643,Q"+

H  + 764H*  SR"$% > "*Eb (($,Sr4* " + f'; /^t,"$% V, 7 < ' E5 $, "3 IXt,Q "'

%Q "$% V,  IM" $,"'q] $"mWIMv7(#6 "A 32 FvQ7"P7>$,"' gEX/f\ 76 fA; / $,"!  o64 $"AH +LP(7 $, " !(# ]X6$,"'5 +3K$, Q"'$MQ7 lQ7 SRK >o,\ 4, tJRK m$,"(#"' V,"'6 F !Y V, k '3KQ7 @ , SRK7>mP >Q]@"T 7$"A  > +.  > ]'< {JQY M F P("  7 W,$,  R"@"r P( " *"g"5rK,"764"A6P\5 7 FY64"'$ƒQ7"

F"IXQ"$,"IM"Q7 SR "XQJ 76l5Y6 64Y F$, 7 IXt,Q64k i#"'$ >v W% V"\ F( 5"A "Q"'$g

  aFf'}]hH3nIXt,Q ($,(# , Y e o a -I"3- fGh EG $ - fX d; "f  7 6 e o a -0" - fGh `/!EG$ -f  m; "7f 9/ j  l4'5$% X V-D a, n fAh*94a jDnp/hJ 76 mY m >  ( P  ,, e"3.@* f794aCPj fAh5v8  'W ;vEM 7 F"A p 6  d; "'f j< h {JQ"P*  N(# ]TQ"$," Y o,Q7 VA3)"R " IXQ  F 7$,$"AH _Q7\6 F$,74   e".7tEG $z J  m; "f 9/ j32IXt,Q Q7& " FIX$,7>  F, V," (7  "    d; "f 9b/ [3Q"vP$,"l> ""'$, ƒEG$,!7   $,$"A F(#76>^ aFf'}]ho F X   VI@ o7 mL7 " $ (#"$EG"'5 F !Y V, v > $,t,Q P"u*"'] + , P( "EG$, Q"*adY67hJ64Y W,$ 4, _E ]"'$"A WAZj$"A5VR "$,+6 V% o,Q"b P"b XIXQ %Qv"'

%Ql7 76 J$,"5,$7"'6.7( l,Q" 6"' V,QlEkQ"b5 $$,"' (#764 >o 764 RK 67 [ "'6v  J"A W, > '< ?"$,"3JI "_Q7 SR "6 V% LE,Q"_EG$,    " M4 #f "i$%$i$ "  "> #f "i$%$i$ " d" K5  3 IXQ"'$"    64"',"' \,Q" Km"'$@E) 7* Y X$," "'

F"A6 XQ"o#"> 77>*ED+"' $ M 76vIXQ "g, >*IJ  J$,"5,$,"'6v _Q"T+ "' $X7(_ ,Q"*"6 E + "' $ 7 R "^,Q7 VToQ7  b64 "'6k V, E ,Q"*( (Y V, p oPPN 7%Q F64"A 

Y b5 \P +v F"'$R "'6vQ V,Q"\\ $F%  tW+v$% V"mEƒR "$,++ >l$%6  !%Q^Q >Q"'$,Q7  ,Q7 VXEG $@ 647t% <  R"'_67 V,*

! O"    _& M  #f "%$i$%$ "  " #f "%$i$%$i" (]34I "b,  Q"T($,46475FZj! PY B e " Q7]46

  

a "  (X " (0" #h3gP

IXQ7"$,"

  

  

X( F 

'(X ( F (



 

] ]

aWf %h

" " 

 0

s `M "

s M jOf#"i$%$i$ "



"

f n0( 3 (XF R fn(XA3 64""A )Q7" e " QK46\"'$  , K "A6mIXQ* 764 RK 67 Y ),Q7 VJ $,"J, > >"A6m74 IXQ7 " % >] ƒ $,"Jƒ$,"5,$7"'6P64$,7>,Q" F76+m 6PX6""' 2,Q"@($,46475ƒEn! PY 7$,P Y , V  7 F, ], 'vC $ (] F L}< }lIM"!% e "m $ FQ v64 F$, 4 LEG$g,Q"P\464"')(7 $% P""$% _"\,Q"  EG$, 6 F$,74 VR"$ Q"J#"Q.PVR "P"] #"5WIM""'.Q7"' "@WIMo"'P("'$, $,"' * S+m#"

Perfect Forward Simulation

25

0.46

0.48

0.50

—

0.52



0.54

0.56

0

0

0

5

5

20

10

10

15

40

20

15

60

25

20

30

80

* 6"."'

F "$TK+^,Q"l ]$,4647H,  EX],"$,\"A64Y V"P,"P(#"$% V7$"A 32I "PEG 76N,Q7 V!!IJ  T7!"A5"A  , $,+

76l 73K l$%64"'$ƒ\* Vr4 \Y "@Q7"b P] Ek P"g (#"]XlQ"WI me"'+m,"P(#"$% V7$"A 3KIM" ]$,K675" Q"'$, X 76.% e k " e  a~d; "'fAh 8e  aF#f ";]h  f 9/4< 07$"'"* p 6N^$,>  aWfA11]/hb5 76475! 5Y  , FY    +K Y bEMQ"A F"l67 V,  76p QVI ,Q7 V! > c  nTf % $ f %4< {JQK7 '3JtEbIM"IX Q  P P ",Q"_Q"> " \"$, vIJ  >N P"^EgQ"(#"$EG"'5l F !7  

> $,t,Q_3Cƒ$,(# t, }< }\ F> >"A W% M, eK > .   c e  a~d; "'fAh  c . F e/aW,f "; h  $@ F"F >P > .@ nTf % $ f %K$,Q 7< }*I "b76 ,Q7 V\IXt,Q .@* c  MEJQ"77  F, "' mEJ,Q"$,"(7Y  7 TQ SR3 " : f QpQ7 mRV  " P PY F"A mQ"v"'>,Q E"'

%Q $7k3  " $P 7 oKm"'$\E@$77 P $,"l$,"'q] $"A6 L>   "'R"  *   Km"'$2E , P( "' EG$  Q"M% $,>"5D64Y W,$ 4, ko,P \> "P"$, g64Y W,$ 4, IXQ_

] P"' _E2 $ 76_/; ;3 ;;;ot,"$% V  
" P"5$,YIJ  >P P"TK+.Q VXEd 5 $'3464"'$"A  7>!,Q"T($,(#$ vE277 k F, "A  l,Q"gQX64Y W,$ 4, K+*$,> Q+\,Q"g , P"b P7 A (# F"'$ $PP"' 7 _ad 6 F, 76 $,6 64"RKY V  ,hoEG$ (XS3 (O 7 6 xI "'$"_d; $ 4f f a~d; $ ;f ]hH3 d; $ - % xa~d; $ ;f'1]h! 76 m; $ /ffTadm; $ ; ;- h $,"' ("AH,R " + 6v $"TP(7 $% 7"g*(7$"'R]  F +.(7Y FQ7"'6_$,"' t% JEG$@  \ Y $@P464"Y '37 F"'"

26

Brooks, Fan and Rosenthal

cM$,Ke4    Ja~/; ;;]hH< 6.3. Example III: Autoregressive Time Series

4 ((# "_Q7 .I "Q7 SR " L7RV $Y V,"_ \"^ "$, "'   "i$'$&$'" YmIXQY%Q IM"#" "'R"  #"^64"A $ "A6 ]+ 

4 $"'>$,"' , F R"mad = hJ($,K"' , MED$%64"$B37[< "< 



IXQ7"$,"   

  U   j 

"



]

a~;d"t o hHb,Q"6 , bR "'H,$  ]73 $"A F(#"'5 R"'+ 3Q7"J7$, F %mRV  "' 76 ,Q" $,"* 76"$ 76  76 % e" (($,(7$Y V,"DEG$,P 'b,Q"5 $$,"' (#764 >b(# F"$, $ 64Y F$,   ƒa R"O7h)IXQ7"$,"37764"$ PK6"m 4 #f "%$&$'$&"t ]3 6   ad~< "< 3     3nQ"'$"ShHP, l($," + $% 64 ($,45"A  ' V,"P($ $A3 ,Q"T(# F"$, $@5 764 7 k64Y W,$ 4,  ƒCa  o  *:; hMY X P\"'!#"$@E,Q"T ]R "$% F"g> PP mEd \  + 6v'  ,Q"$,"5EG $"g#"o  \(7"A664 $,"'H,+lEG$  , X F, V, 7 $,+.64Y W,$ 4, k "! 764"'("'764"'   a~d; "t o h($ $% XEG $"A %Q^EƒQ7"\ 4 $"'>$,"' , F R"T(7 $, P"5"'$, 764"'$g K+ P464"' 6 ($ (] F"*7(#6  >^,Q"$,$"' oP464"' (7 $, \""'$, T F >  7 m W,"(7 mpIXQ7 %QpIM".(n6 V," ,Q"T"$,$,$JRV $, 75"EG$  , @ F, 76 $%6v564t, 7 k6 F$,74 _ 76,Q"vQ7"o 4,$,">$,"' , R "@(7 $% P""$%

 l4%e_(n6 V,"oEG$, Q7" $ UW ]T564t, 7 264Y W,$ 4,  ad "" ƒQ "$%  76^cM$,] e4 /; ;fAh5Pl\VR "bl K+lP464"0  =!Am; "i$'$&$'"O  Z(!IXQ"'q]7  (7$ 7   tW+ lEG$, ,Q"P($ $A< ?VI "'R"'$'3tE   ;73,Q"I "\64$, SI l"I RV  "mE  o EG$  ,Q"m$,$"A F(#647>*EG 2764  [3 AF Ca  o h5< {JQ7"k3K+lC $ (] F  4< fT 76lQ"bEd

HX,Q7 V BE,Q"TPK6"Y   $,"b"' F"A6B3IM"gQ SR"Q7  '  P   f," a    ca Ah cAh c  "  IXQ7"$," 2ca 7hD64"',"' Q"X(7$ $2($,7  W+o  , F45Y V,"'6TIXQPP464" \ 76   Y Q"X e " Q7]46!"RV  7 V,"'6

JQ"o y @&





o 

   

|

     a    h  9 

o |X 76

]

  a  h 

   9  o $

7. Discussion

Ÿi N,Q b( (#"$A3kI "* ]$,46475"* v"'I  > $Q7 ad@ > $Q7 7< } hEG $T($4645 >(#"$EG"'5!  P("A gEG$ s

% $,>"\64Y W,$ 4, : ƒaOA h5< {JQ"  > $Q7 Y P R "$% F pEbM0D{XC  F > F !Y V,"'6 "P(#"$,7>NLMN3

Perfect Forward Simulation

27

bKR"'$F,"'6vv .EG $IJ $%6KZ[ P"o  >$,Q 7 7>{JQ"'$," /4< fo,Q"b57"'5 l#"5WIM""'*Q7"b  !Y V,"'6P,"P("'$ >P 76l$"'R"'$, 7" UWP( LML ,%Q"'\"A a~@ > $Q < / hH3ƒ 6 $,"Y V"$\(#"$EG"'H* m   ,%Q"'\"lLQ"Q7"$,+ E $,> $,7  76 J v $,eVRP%Q7  KR"'$> "7"$% V"A *Q7"!5 "'5 vP,Q"o* Vr4 !  e " QK46vRV  "3# 76Q7" (# , F  W+oE#64"$Z["A W,*  > 'A *$"A 26 V% . F"'*, ,Q""u*"'] P( "P"' % V, .EB7$ P"5,Q46 76PK+\P(7 $ >T7$ ($,(# "'6P m  . ,%Q"P"A )IXt,Q ,Q"T$"FUW"'5 _  \(7 >PP"5Q7K6k< {JQ"m F"@ED"'\(#"$% V,$,"' '35 ! "A6IXQ^Q"'$g  , F45Y V,"'6I "'> Q], '364""$,P7"TQ"!($ 7 7 tW+lQ7 

.  > "o$,EDQ7"o("'$FEG"AHb F !Y V, ^ %Q7"P"o64"' ,5$, "A6_> $,t,Q 7< }P($VRKY64"'  *$"A   , V, lEG$, ,Q"!5  6_64Y F$, k< ' KR]  F +3IM"TIMY6v e"b,.* e"g,Q ($,7  W+  @Q> Q  X(# , F  "o 76v,QY '  "^

%Q"'R"A6 K+ Q7 SRK >N

* F* @ "5*E"P(#"$% V,$"A *  P(# , F  "< ij* *"A $PEG$, "Aq]7 V, 7 a %h 76 aWf KhX,Q7 VTIXQ"' %  wf3#Q7"\RV "\E "(7  L b64""'$P "'6"] $"'+K+_,Q"!,$, 7 t, ^EG$,

7> ","P("'$, $,@" ls   c < 4((# ",Q7 VMQ"gP7PY F >\RS 7" LaFf ]h Y Ca "O7h  a    "t   h534 FmQ7  ') e/ca    "O c Oh Va    "tBt|    h5"'P"5,$Yl6 F$,74  EG$ "o! ,%Q"'\"7 F >\64$, SI@ )EG$  ,Q"QM64Y F$,  ,*4%  v$,"' Y , V  MEG$  Q"m5Y6v" < {JQ"o> ""'$, B$"FUW"'5 _  \(7 >. ,%Q"P"T($,45""A6 X  XEG VI@ '<  R"'v P64"'7 FW+ NEG$  IXQY%QIM"bIX QvP,  _  \(7"A 34IM"T v 7 W,"'

664$% SIO$,"' Y   7  "i$'$&$'"t  EG$, P"lQ"'$m64"'7 FW+  76

5"'(4   g  \ ^$,"' Y   NEG$, N IXt,Q ($,7  W + NDa  [*h 9;  nCa  ~h  IXQ"'$ "   (  NDa ƒ*h 9 Ba )hH^ ,%Q"'\"* F >EG$,z64$, SI@ g,^  P("PEG$, Q7"*", 7a  "5hT6"7 tW+ Q7

 ^ '5"(, 5"o$% V,"oE PV "!$,"' Y , V  EG$, N^Q7  X \> "P"$, g64Y W,$ 4, IXQ_P"' Lf 9 PVb,Q"@(#"$EG"'5 F !Y V, * %Q"'P""$g#"m7 "'6L  b .$,"WUW"AH, L , P( >_64"'7 FW+3k  75"   O  o .a -I"2O| m"O f h  -j 2 ,%Q"P" KK6 %Q  "E,Q"' "_]"'$P"A64 "v"'P("'$, $,"' a~ 76 5 $$,"' ( 764 > 64 F$, 4  ,h\5 Y6 #" "rK$,"P" +*Q" (4EG[ Q ,+h ^Q7 bP$,"\>"7"$%  '  "32IXQY%Q * S+L P!,Q"l(#"$EG"'5\ m   > $,t,Q0   T F"EG7"' , o  ($% H, ".] M\ $"l"5u.5 "] $,"' , X' v"o64"'R"' ("A6l vQY @ $,"' 7
] Y F"'$!CD"5,$] "  ( $F%  \ 76   ,Q"v(7 $Y5 (7 ]% !  ,Q" L {X = 9L $e4 Q(P lLML 464"'B Q  "3]vK(", "' '3 $,""A5" 3  l@7>7 FJ/;;f 3VEG$  F( $% V *$,"Y V,"'6 ,\QY X(7 ("'$' $, "EGk Y F\, "'Y iL %Q7  '3 "A F(#"$@  "'$@ 76v9x tEG$, 6 hb"'76  k  BEDIXQ  IM"$,"JeK76P"'7>Qm!64Y ,57 , ƒ 6\PP"]ƒ\,Q"@7

FYM 64"A  )7764"$,(  >T7$ƒI  $en"A k,Q" ((#$)E#Q"  h  > ""'$ >b 76PC Q]+4  ' 74"'75"A = "' "' $,%Q* 5 764"'$ƒ>$% ] Km"'$@0 V; ; } %K"A ((#$JEG$    =  EƒM 7 6 7

W,7> M@ >$,Q_ "7"@$, "A ƒEG $M^ $e VR %Q7  v^]"mM $ \ > $,t,Q* L  >$,Qs^> ""'$,  %Q7  7 '<   `d   f e&,e;f g    Edbf    g 2  3ƒ/1; }fA0$,""'* k3nn  \"'7  k"'75" = V,"' 2EG $ƒ^ $,eVRm Q7 P  ," M $,< dYe;f  d #   f .&    $g  .&  56gSgd &   d  3#!8  -=  "],Q7 [3 "7"@,. W% V  $,tW+ < K"A5 76B3> $,t,Q 7< }Q  2P5 P\ !IXQ\RV $, 7 KR"'$> "7"ƒ$% V"J F764 "' Xad"< >7< = ] F"' ,Q7 nfA11  3 =  "]Q7 Jf'11 -3  VIX "' g 76 =  "],Q7 Jf'1 187'3 7   J/; ; /hXQ"P"'"'65 P(4"* P $Y , V  ( $% P"5,"$ 'A<  $  B"A%QYq]"' aG#Q\ 7 +], ƒ 76TKP"$,Y  hn"'\(7V+ "'6g5P(" 'EG $5 KR"$,>"'75" $% V,"\ F64"A 3k'  Y F"P"'\(7V+ "'6, P( "P"]o@ > $Q < }7"X .,Q7 VJ7$ 'M!7 FJI  $emEG$   # F, "' J"g 64RV  % > "@Q VMQ"b$,"' t,7>m  > $Q7v3] e ",Q"$,"'  >P5KR "$,>"5"5Zj$% V"J$,"' , M"'VIT3KY m7%Ql"A "'$'3]J$,"'q] $ >o" < > 7 "5A $,t,Q < }7< {m#"> v7$M F76+*E2KR"'$> "7"X$, "g"' F * V  3]IM"b64"577",Q" d    Yf    d  $g   '& M"WI "'" WIMP($,7  W+*\"A  $,"' 6ƒBa A¢hJ  6 )BOa A hX_ \ F, "o (7 " _3IXD Q #Zj > "$%  34K+

6ƒaOA h n )BaOA h

6

) 

 6 F7( ƒaCJhn BaCJh Q$ ; 

9N"o Q7  B"T( $Y5Y $,+* ]"'$"A W,"'6_#76>P,Q"o64Y W% 7" 1

a"t7hQ"%A nE ƒaOA h   "

IXQ7"$," 1! aC"O "iA¢hTY T,Q"64Y F$, pEJ,Q"%Q7  E "'$!%p F"( oIXQ"' W% $"'6p a"t7hH3ƒ 76p ƒaBA¢hTY P"T F, V, 7 $,+.64Y W,$ 4, k<   ]R "$,>"'75"g$, "T$,"' , @ $"b(# , F  "TtE)7$@ F*   "5@Y )UW7 F P 7 "5E,Q"! F, V,"o F( 5" 34X,Q"+ $,"'q] $,",Q"b7 F"bE64$,tE 764  M 76\ $"b5 P(Y "A6*Q7"$,"* bad ""b" < >P5 7 W,$H *$,"5EG"'$$,"'6m,!"' $  "$A|] ""@" < >7<  m 7>PC $ (] FZ N , _!
"'75"g P"3k64 RK 6"'6^K+ % S3n #76"'6

VR "K+.    M\Pa ' h $, 764  RV $Y  " $Q7 7< }< iŸ764""A6B3 @ >$,Q 7< } 3 LH  , 'FZŸ P [3

76  c Y J~< ~< 6B< Z[  e"3 FPIM"g 4, Q7"bEG VIX >7
 >*Q7"o764"'$ +K7>. F !Y V,"'6v,"P("'$ >l%Q  k3,Q"$,"o* S+l#"m P"bEG$,""'6  6"5"'$P "TQVI`!7%QvE2Q"b,P"b,Q"o%Q7  vIX  ("'76vvQ7"TRS $  M7 F7"]X"'\(#"$% V,$,"' '"'$ IM" ' v* e" ' g 6.,Q"o F*  "$XIM"T _* e " %ZV"'75"$, "b>"J#"5"'$ ,Q"oP$,"T e " +.Q7"!%Q7  _ @bUWP(, " c < ' v,Q"o,Q"$Q7 6B34Q7"!P$,"b e"'+Q"m%Q7  _ J,TUWP( , " c 3Q"o "' , X,P"!Q"!%Q  IX 2 (#"76_   Nqb ^ W% V  $,tW+  F, V, 7 $,+L6 F$,74 k $,t,Q 7< }3n 76Q"\7 "5EG "' , E)Q7"m$,"'  >l , P( "' EG$  F,  7 $W+3KIXQY%Q_I "b"rK(7 $"EG$Q"'$@v,Q"o5  ,"5rKXEQ"o P( "T"5r \(7"b 4"'H, -< f< (*),+.-0/1  4