Úvod – motivaˇcní prehled

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Úvod – motivaˇcní prehled
Kapitola 1
Úvod – motivační přehled
1.1
Intuitivní smysl pojmu PN
• !"#$%& A, B '()*+ ,-.
,"7-5-)(.8
!"# !"$%&' (#)*$%# '('&/0 1(23)&!%"!4& '( 13!%(5#'6/6
P (A ∩ B) = P (A) · P (B) ,
,9&:(;7 $*"+$"$,*- . /.0 '(+ 7( 12!,(314(45*4."
*!."'*- 6#,+ B ! *!41!&<
*!."'*- 6#,+ A *#*- 4,0$,*3*!
Příklad (dvě mince). =3!%(5#'6/6 5)- '()* '!"# 29(';> 1(23)&!%?8
'() A = 1- ,@)16 ;&1/& ,-51( "@(% ,
'() B = 1- [email protected]#0? ;&1/& ,-51( 0%-)- .
• A")>4B&1"# 13! 1(2-'6;3 1(23)&!%"!4 '(51"4%&)C/0 '()D+ -%( !,6B(
+ .4(@? 4>;4" '()D; "5,")65-'6<
+('04."-
*#7',$.04."
Příklad (dvě mince podruhé). A96.%-5 ":&)&51> 1(23)&!%C/0 #53%"!468
#53%"!4 a = /" ,-51( 1- ,@)16 ;&1/& ,
#53%"!4 b = /" ,-51( 1- [email protected]#0? ;&1/& .
• E*4" #53%"!4& '!"# 2,@-)&5%- ;-4(;-4&/.* ;"5(%")31* ,";"/6 130"51C/0 )($
%&:&1+ /"7 )(5( . ,"';#
+ ,"4-7;" ,-. . ,"';#
1(23)&!%"!4& σ $-%F(G(@<
*#7',$.04."$ *'84(*9%8 ,#0$:$*
H14#&4&)16 !;*!% ,"';# A= '( ,"5"G1C+ -%( 1-)6/ 4# ;3;( ,"5;&I#'6/6 '()*+
@(!,(.4&)( ,"5;&I#'6/6 130"51? )(%&:&1*<
Příklad.
5# 1- "G>5 5" @(!4-#@-/( - 2-'6;-'6 ;1( 49& #53%"!4& ) :-!")?; !%(5#8
a <<< /" !& "G'(513; . ,&46 J1- )CG>@ '( PIVO - VÍNOK
c <<< /" !& "G'(513; . 0%-)16;# '65%# J1- )CG>@ '( VEPŘOVÁ - RYBAK
b <<< /" !& "G'(513; '-." 5(2(@4 J1- )CG>@ '( SALÁT - ZÁKUSEKK
1 Úvod – motivační přehled
4
!"# $%&'&()* )"'&+%&,)-. / 0'&1& 0&23456# 7/8& 0&#9$82 0:; '&+%&,&56)4 %6<
+")4 #;)$4= > 0:40/,* 0'5)4 2,6?&!1; a !; %&,4# 7",)&,[email protected]" #;)$4= > 0:40/,* ,'2%A
2,6?&!1; c !" /?" ,&#)456#. 3" 5"0:&56 !" %&,4 54$ 8 0;52 / 'B(/ 54$ 8 54)2= C?"
)"7!"# !; +$"?/ 7;!1. 1/8 !; &0*1 0&#932 %6+")4# #;)$4. 1")1&8'61" ,5&2= D&+%&<
,&5/$4 /?E&';1#2! (2," )6!?",274$4= > 0:40/,*. 3" #6# &(7",)6)& 0;5& / )"0/,)"
2× &'"?. +5&?4# 5"0:&5&2. 0/,)"<?; 2× &'"?. +5&?4# 'B(2= F&,&()*. 7"!1?;3" #6#
&(7",)6)& 54)& / )"0/,)" 2× &'"?. +5&?4# 'B(2. 0/,)"<?; 2× &'"?. +5&?4# 5"0:&5&2=
G& !" 1-H" 1:"14 2,6?&!1; b. +5&?4# /)/?&E;$8- 0&!120= I&#)456# !". 3" 5"0:&56 7"
12H)6 / )")4 5%&,)A 7; ,&0?);1 !?/,8-# 8/?&';$8-# +682!8"#. /?" [email protected]" !/?61"#
! &(!/%"# 5;1/#4)9= J/&0/8 'B(/ 7" +,'/56. #6?& 8/?&';$86 / $%B(*74$4 8/?&';" 7"
5%&,)A ,&0?);1 +682!8"#= F'& 8&)8'A1)4 '&+%&,)214 !; &0*1 %&,4# ,5*#/ #;)$"#;=
G"?8&5A 0'/5,*0&,&()&!1; 5B0/,/74 )6!?",&5)*K
1
2
PIVO
3
4
-
VEPŘ.
1
4
4
VÍNO
3
4
-
SALÁT
1
4
H
*
HH
H
HH
j
1
1
2
3
4
RYBA
H
*
HH
H
HH
j
1
4
3
4
ZÁKUS.
L 0&0;!2 !;12/$" 7" 0/1')A. 3" 5 &8/#3;82. 8,B ,&74,6# %?/5)4 74,?&. 1",B +)6#
5-!?","8 2,6?&!1; c. 7" '&+%&,)214 & 2,6?&!1; b )"+65;!?A )/ 1&#. 7/8 ,&0/,?/ 2,6?&!1
a= M& 0'65* 5B!1;%27" !;12/$;. 8,B !"#$%&' a ( b )%$ *$!+,-.-. -/0"1'%#2 !"-$
3$!-$&(+' !"#$%&' c=
M")1& 0:48?/, '&5)*3 ;?2!1'27" ;)12;1;5)4 '&+,4? #"+; 0&,#4)*)&2 / )"0&,#4)*<
)&2 )"+65;!?&!14 2,6?&!14= >!82182. 0&+&'27"#" [email protected]@4 8&'"?/$; #"+; 0;5"# / !/?61"#.
'"!0"81;5" #"+; 54)"# / +682!8"#K
)
9
(PIVO, VEPŘ., SALÁT) → 12 · 43 · 34 = 32
10
(PIVO, SALÁT)
→ 32
1 1 1
1
(PIVO, RYBA, SALÁT) → 2 · 4 · 4 = 32
)
3
(PIVO, VEPŘ., ZÁKUS.) → 12 · 34 · 14 = 32
6
→ 32
(PIVO, ZÁKUS.)
1 1 3
3
(PIVO, RYBA, ZÁKUS.) → 2 · 4 · 4 = 32
)
3
(VÍNO, VEPŘ., SALÁT) → 12 · 14 · 43 = 32
6
(VÍNO, SALÁT)
→ 32
3
(VÍNO, RYBA, SALÁT) → 21 · 34 · 41 = 32
)
1
(VÍNO, VEPŘ., ZÁKUS.) → 21 · 14 · 14 = 32
→ 10
(VÍNO, ZÁKUS.)
9
32
(VÍNO, RYBA, ZÁKUS.) → 21 · 34 · 34 = 32
4-& '&'1-, %+5%# *$)+ 67 7" 1",B )6!?",274$4K *8' 0-(#$%&' 3$!-$& *$!+'9 ),:, 1/;
#'<'-5 c %/ 3$!-$&5 1(=$1(->:3 1/#'<'- a ( b -(10")/+ -/$1#'19 ),= F'& N&'#6?)4
+60;! 1/8&5A !;12/$" (2,"#" 0&2345/1 !B#(&? a ⊥⊥ b | c=
• O/1"#/1;$8B (B$%&# 1& #&%?; 5B76,:;1 !$%"#/1;$8B 5 1"'#4)"$% 0&,#4)*<
)-$% %2!1&1K
p(a, b|c) = p(a|c) · p(b|c) .
1.2 Kde se vyskytuje pojem PN
5
! "#$%& '(%') *+!$'&"# $*!,#-./01%2 ,/ "! %/ */345'&'6 7($%/%12 ,/8 9(3!
(:#'&'/ 0 31;<54 %!=%(> *?/ @3! (018/01'A !;!4!'%6?'5 3#$.?)%'5 *+5*13 ./'B
.?)%'& 0 /335;! C DEFE G %/4%/ 4/%#01:'54 H0/3( 0<1. */3?/9'/$%# @64&?'&
0I'!,76064E
• J/3$%1%') "!2 8! JK ;@! ,76*1% "1./8%/ !"#$%"%$!&' ()*+,2 0I$%#7("5,5 (?:#%)
[email protected]%17I [email protected]# '67/3'A4# 0!;#:#'14#E
• #'6 4/8'6 #'%!?*?!%1,! [email protected]%17( JK $/(0#$5 $ */"4!4 -" %).$/"0+ L$3?(8!')M
7($%/%IE N.0#01;!'%'& ;@! JK (36;/$%5 a 1 b @1 */345'.I (36;/$%# c O/?4(;/01%
,/9I */81310!.2 8! 7($%/%1 L0 @3! (018/01')4 3#$.?)%'54 *+5*13& $! "!3'6 /
7($%/%( 0P:# 1?#%4!%#,.) 45+!2 '&.3I %)8 '[email protected]') :5%1,5 45?1M $! O1.%/?#@("!
'1 30& $;/8.I2 *+#:!48 "!3'1 @ '#,7 @60#$5 '1 (36;/$%!,7 a 1 c2 @1%54,/ 3?(76
'1 (36;/$%!,7 b 1 c Q
p(a, b, c) = f (a, c) · g(b, c) .
• R/ 46 '6$;!3("5,5 [email protected]'14E S9I,7/4 0 3#$.?)%'54 *+5*13& (,7/01;# 0 *14&%#
*/:5%1:! %+5?/@4&?'/( 7($%/%(2 $%1:5 (,7/01% 301 30/(?/@4&?') */%!',#6;I
LT *14&U/06 */;! *?/ 7/3'/%I f 1 g M2 ,/8 1+%2' (",34)!5 &6.) 7 E
1.2
Kde se vyskytuje pojem PN
! '&./;#. /9;1$%5 41%!41%#.I2 0 .%!?A,7 '[email protected]! '1 */"!4 JKE GI$.I%("! $!
'1*+5.;13 0 %!/?## *?103&*/3/9'/$%#2 ./'.?)%'&2 0 '67/3'A,7 *?/,!$!,7E K1*+5.;13
41?./0$.) *?/,!$I "$/( 3!-'/06'I $ */4/,5 %/7/%/ */"4(E G %/4%/ 4/%#01:'54
H0/3( 1;! @45'54 "#') %+# /9;1$%#2 .%!?A,7 JK 7?1"! */3$%1%'/( 1 3P;!8#%/( ?/;#E
!3'6 $!Q
• / %!/?## ./'%#'V!':'5,7 %19(;!.2
• / 4'/7/?/@4&?'/( $%1%#$%#,./( 1'1;[email protected](2
• / *?103&*/3/9'/$%'5 !=*!?%'5 $I$%)4IE
W/360642 8! JK ,/9I .01;#%1%#0'5 */"!4 ,71?1.%!?#@("5,5 [email protected]%17I [email protected]# 0!;#:#'14#
$! ?/0'&8 0I$.I%("! ,$,) )8#"9% %+).$+ (."!:3():)8&)9%$ " 9%"%$9%$ 7; 1./ %1./0A
'1<!; (*;1%'&'5 0 '!"[email protected]'&"<5,7 .1;.(;!,7 *?/ */*#$ @'1;/$%5 1 '!"#$%/%I 0 (4&;)
#'%!;#V!',#E
1.2.1 Kontingenční tabulky
• !3'6 $! / /9;1$% 3#$.?)%'5 $%1%#$%#.I2 .%!?6 $! @19A06 "%+<).$6#&',$ :"%7E
• K6$%#' (018/01') $#%(1,!Q
N EEE '!*[email protected]'6 ./'!:'6 4'/8#'1 #"9$= ">&'0? .$%5.$'2
∀ i ∈ N Xi EEE '!*[email protected]'6 ./'!:'6 4'/8#'1 @.)!&' #"9$= "0+E
• W1%1 "$/( */$;/(*'/$%#
Q /9"!.%P ?/@%+53&'A,7 3/ *+5$;(<'A,7 8A&3 2 %"E *?0.P
.1?%)@$.)7/ $/(:#'( i∈N Xi ≡ XN E
Konvence 1.1 (značení). !B;# x ∈ XN , A ⊆ N *1. $I49/; xA /@'1:("! ?!$%?#.,#
Q
:# *?/"!.,# x '1 XA ≡ i∈A Xi E
1 Úvod – motivační přehled
6
! "#$%!&' &!( )* )*)(!+, !"#$"%&"'"( #)*+, )- ./01#%2' "/30!"*2,- $(*04
$!5&4 3678* x ∈ XN 9:;:!&, 9/<*( /3=*$(> n(x) "!:!"*[email protected] &/ (4(/ 367$AB
• C(!(;)(;8; +'(D;2/6 &!(! 9/+!56=, "! 2#@/&2? +?3'0 " 0/"&'%*2,- $(*04 =* 60<*2/
@6)(/(/6 p +><; !0;(1*(;8$4 1,:* 2! XN ! @%*&!=, 2'=!$? 1/&*% 90/ +A)+'(%*2,
&!(B
• E3%,3*2/6 (:,&/6 1/&*%> =)/6 (!$"+!24 ,!%-,$"&./"( 0!1&,2B F:*&9/$%#&# )*5* (*/0*(;8$# @6)(/(! p =* )(0;$(2' $%!&2#B F/(/1 =; %"* "%/G!0;(1/+!( ! "!
60<;([email protected] )(!2&!0&;"!<2,[email protected] $/2+*28, 2!9)!( =*&2/"2!<2? 0/"$%!&
•
ln p(x) =
X
λA (xA )
90/
x ∈ XN ,
A⊆N
$&* 2*"#9/024 .62$8* λA )* 2!"?+!=, $"#&/) 3&B H!(/ (*01;2/%/G;* 9!(02' 9/I
[email protected]#", "* )(!(;)(;8$4 .A";$AB
• J*&2/(%;+4 $/2$04(2, %/GI%;2*#02, 1/&*%A )* 60<, 9/5!&!+$*1- 5* 2'$(*04 ;2I
(*0!$8* )* 26(2' 26%6=,B K (:,&A 4$&/)/34$3 534 ,!%-,$"&./"(34 0!1&,6 =* )/6I
3/0 ('[email protected] ;2(*0!$8,- $(*04 )* 2*16), 26(2' 26%/+!(- 60<*2 &'&;<2?1 )A)(41*1
9/&12/5;2 N B LM'&;<2?1 )A)(41 =* (!$/+?- 5* ) 12/5;2/6 /3)[email protected]=* =*=, 9/&I
12/5;2ABN H/ =*)(- 26%6=, )* ;2(*0!$8* +ADD,[email protected] :!&>B
• *=9/96%#02'=D, (:,&/6 1/&*%> =)/6 %/)73 8 ,!%-,$"&./"( 0!1&,2 - (=B @;*0!0I
[email protected];8$4 %/GI%;2*#02, 1/&*%A- 6 2;[email protected] )/63/0 2* 26(2' )* 26%6=,8,[email protected] ;2(*0!$8, /&I
9/+,&# )A)(416 O9%[email protected] 12/5;2 2'[email protected]/ 2*/0;*2(/[email protected]/ G0!.6 ) 12/5;2/6
6"%> N B LP(*2#: 2*/3*"2#1*2? ) ('1;(/ G0!Q8$?1; 9/=1A 2!%*"2* 9:,)%6D24
&*Q2;8* + R SBTBN
• H,1(/ "9>)/3*1 $!5&? 2*/0;*2(/+!2? G0!. 60<6=* 2'=!$? 9#)#$9#$3 5 0!1&,
+A)+'(%6=,8, &!(!- (=B 2'=!$/6 12/5;26 90!+&'9/&/32/)(2,[email protected] 0/"&'%*2, 2! XN
+A1*"*2/6 )9*8;Q8$?1 9/5!&!+$*1B F/&)(!(24 =*- 5* G0!Q8$4 %/GI%;2*#02,
1/&*%A %"* [email protected]#9!( =!$/ )(!(;)(;8$4 0!1&,2 9#/+ #+/ :;B H/ "2!<,- 5* $!5&?
(!$/+? )(!(;)(;8$? 1/&*% 1>5* 3?( *$+;+!%*2(2' "!+*&*2 =!$/ (:,&! )(0;$(2'
$%!&[email protected] 90!+&'9/&/32/)(2,[email protected] 0/"&'%*2, 2! XN - $(*04 [email protected]/+6=, 60<;(?1 +"(!I
@>1 F B ;8142'- + (/1(/ 9:*@%*&6 2*9>=&6 &/ &*(!;%>B
1.2.2 Mnohorozměrná statistická analýza
J*&2# )* / /3%!)( )(!(;)(;$A- $(*0# )* "!3?+# +"([email protected] 1*"; )9/=;(?1; 2#@/&2?1;
+*%;<;2!1;- "90!+;&%! G!6))/+)$?1;B
• F0/ 9/9;) +"([email protected]> 1*"; +*%;<;2!1; )* 9/65,+!=, ("+B 0!1&,2 9#/+ #+/.,"(34 /!<"$3- !2G%;8$A 9#/+3#+/), &=+)#$!" 0!1&,9 L>[email protected] J)/6 (/ )A)(41A .62$8;/2#%I
2,[email protected] +"([email protected]>- "90!+;&%! %;2*#02,[email protected] +"([email protected]> 1*"; 2#@/&2?1; +*%;<;2!1;B HA9;8$A
$!5&# +*%;<;2! =* +A=#&:*2! 9/1/8, /)(!(2,[email protected] ! )+4 +%!)(2, [email protected]/+4 2#@/&24
+*%;<;2AB
• F!(02' 2*==*&2/&6DD, (:,&/6 1/&*%> + (4(/ /3%!)(; =)/6 ("+B /& +/A$<"( 0!1&,2$(*04 %"* 9/9)!( ) 9/1/8, !8A$%;[email protected] /0;*2(/[email protected] G0!.>B H!$/+? 0*$60";+2,
1/&*% =* "!&#2 9:;3%;52' 2#)%*&/+2'B
•
1.2 Kde se vyskytuje pojem PN
7
!"# $%&'() *+),"-. -/0121() ε1 , . . . , εn 3 45/6. 7!"# (%-$'7/8 (/$'-1!0.3 9%#!:
!"-!4. ! (#0"-"# !5;/&(< +"&("5"# % (/(#0"-=81 6"$>5)0) σ1 , . . . , σn ? @-%A#:
7/8/ !)!5.8 6"-(1*
x 1 = ε1 ,
x2 = α2.1 · x1 + ε2 ,
···
xk = αk.1 · x1 + . . . + αk.k−1 · xk−1 + εk ,
···
xn = αn.1 · x1 + . . . . . . + αn.n−1 · xn−1 + εn ,
4&/ "-B/8 (C45/6. $ 4"/D*1/(5E αk.i 8"+"# ,=5 -)(/*+'()? F%43 >6" 01,"-"0:
("# DG(< -"0,# $,)0=*+ HI (/-)(/*+%(=*+J 4"/D*1/(5E3 +"&("5) *+),"-=*+
-/0121( 7/&("$(%2(C #62#7< +"&("5) !0"A/4 ('+"&(.+" -/45"6# [x1 , . . . , xn ]
*",) ;/B/(< 5"+"5" !)!5.8# 6"-(1*?
• K<!4%(= ('+"&(= -/45"6 8' 6/9#0'6(< 9%#!!"-!4. 6"$&C0/(<? L;<&# -B/*+ 5C*+5"
6"$&C0/(<3 >%6%8/561$"-%(=*+ "(C81 (/(#0"-=81 6"$>5)0) σ1 , . . . , σn % M(/-):
(/*+%(=81N 4"/D*1/(5) αk.i 3 0$/ *+'>%5 7%4" !"!# !#$%& '()*+?
• O>C53 >"&!5%5(. 7/ >"$"6"-'(<3 A/ 5#5" 5;<&# 6"$&C0/(< 0$/ *+%6%45/61$"-%5
! >"8"*< >"78# FP? L" 7/!53 0$/ 71 *+'>%5 7%4" !"!# !#$%& '()*+ !,-%!-,.
/03 - 5"85" >;<>%&C 7%4" 5;<&# 6/9#0'6(<*+ 9%#!!"-!4=*+ 6"$&C0/(<3 45/6.
-)+"-#7< #6215=8 -$5%+E8 FP?
• L%4. - 5.5" !15#%*1 +6%7< &E0/A15"# >"8"*("# Q0"+# 96%R)? S%A&.8# 6/4#6:
$1-(<8# !)!5.8# 6"-(1* 0$/ >;1;%&15 (C7%4= %*)401*4= "61/(5"-%(= 96%R? @$0)
96%R# "&>"-<&%7< ('+"&(=8 -/0121('83 8EA/8/ 7/ 5/&) $5"5"A(15 ! 71A #!>";':
&%(=81 1(&/G) 1, . . . , n? T" k :5.+" #$0# (%46/!0<8/ B1>4# $ i:5.+" >;/&*+"$<+"
#$0# >6'-C 5/+&)3 7/!501A/ 4"/D*1/(5 αk.i (/(< - !)!5.8# 6"-(1* -)(/*+'(U
i→k
⇔
[ i < k & αk.i 6= 0 ] .
• V=B/ $8<(C(. -$5%+) FP H#62#7<*< "(/( !5%51!51*4= 8"&/0 *",) 8"&/0 !5#4:
5#6) FPJ 0$/ >%4 &/5/4"-%5 ! >"8"*< 71!5.+" 96%D*4.+" 4615.61% (% $'40%&C
5"+"5" %*)401*4.+" "61/(5"-%(.+" 96%R#? /&(' !/ " #6215"# -/6$1 M#!8C6(C(.:
+"N !/>%6%2(<+" 4615.61%?
1.2.3 Pravděpodobnostní expertní systémy
• /&(' !/ " ",0%!5 #8C0. 1(5/019/(*/3 - (<A !/ 6"$+"&"-'(< $% (/71!5"5) &C7/ (%
$'40%&C 8%5/8%51*4=*+ 8"&/0E $%0"A/(=*+ (% 5/"611 >6%-&C>"&",("!51?
• P'!51( #-%A"-%(. !15#%*/U
N ??? (/>6'$&(' 4"(/2(' 8("A1(% 1"%!(,2 H!)8>5"8E 21 6"$+"&(#5<J3
∀i ∈ N Xi ??? (/>6'$&(' 4"(/2(' '3(4#3" 5()3(! i:5.+" R%45"6#?
• W1&B5< /G>/651 8"+"# >">1!"-%5 !-. $(%0"!51 " -$5%$<*+ 8/$1 7/&("501-=81 R%4:
5"6) -/ R"68C $-0'B5(<*+ 18>014%*<3 (%$=-%(=*+ ()6(7(6"$8 9,"6#)+"?
Příklad. X#Y5/ xi ∈ Xi 3 i = 1, 2, 33 4"(46.5(< +"&("5) R%45"6E? L%4"-. "&-":
$"-%*< >6%-1&0" 8EA/ 8<5 5;/,% ('!0/&#7<*< 5-%6U
1 Úvod – motivační přehled
8
!"#$%&! X1 = x1 & X2 = x2 , '() X3 = x3 .
!"# $%$&'() &*+,'-# .# +/$*-0) $1% #2$#1+(3 &/&+45/ $1'-,.3-3 6#7 (#.*&+%+/8
9'0%:! #2$#1+(3 &/&+45 .# $%;3+';%:! $1%<1'5= 0+#1! (' 7)0>'[email protected] :&+,$(3 *(A
B%15'-# :# B%[email protected] &%,C1(, %?:%7%:'-3-C $1':*?#> 5) ('.3+ %?$%:@D (' ?%+'7=
-% >7# :/:%?*+= $%0,? ()5 (@0?% 7'?) ,1;*+4 0%(014+(3 C%?(%+/ B'0+%1E8 F#
.'&(4= G# : #2$#1+(3-C &/&+45#-C $1'-,.3-3-C 6#7 (#.*&+%+/ &# :/,G3:'.3 $H3A
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9
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10
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11
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15
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Význačné konference a zahraniční skupiny
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16
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