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$//1*=&>,$'1 yi '1--& 5$*0&
P(Yi = Cj| −i,!, β, k) =
exp
(
β∑
l∼kj
ICj(yl)/
Nk
)
G∑
g=1
exp
(
β∑
l∼kj
ICg(yl)/
Nk
)
?@7AB
#$=1 Nk *&))*1/1'%& -& #,01'/,$'1 #1, k =,+,', ),C )*$//,0, +$'/,#1*&%,9
β > 09 y−i = (y1, . . . , yi−1, yi+1, . . . , yn) 1 ICj
(yl) ,'#,+& -D&))&*%1'1'>& #1-2
-D$//1*=&>,$'1 yl &--& +-&//1 Cj9 +E1 #,)1'#1 #&--& #,/%&'>& 1(+-,#1& &--D,'%1*'$
#1--1 +$=&*,&%1 X = {x1, . . . , xn}7" #,F1*1'>& #1- 0$#1--$ -$.,/%,+$ %*&#,>,$'&-19 G(1/%D(-%,0$ H )&*&012
%*,>>&%$ #& (β, k) 1# H #13',%$ /$-$ #&--1 /(1 #,/%*,;(>,$', +$'#,>,$'&%19 +E1
=1'.$'$ (%,-,>>&%1 &'+E1 )1* -& )*1=,/,$'1 #1--& +-&//1 #, ('& '($=& $//1*=&2
>,$'1 yn+19 ('& =$-%& '$%$ ,- *1-&%,=$ =1%%$*1 #, +$=&*,&%1 xn+17 I- )&*&01%*$
β /1*=1 )1* 0,/(*&*1 ,- .*&#$ #, ,'J(1'>& #1--& +-&//1 ),C *&))*1/1'%&%& '1,
=,+,',9 #&- 0$01'%$ +E1 β = 0 +$**,/)$'#1 & ('& #,/%*,;(>,$'1 (',5$*01 )1*
%(%%1 -1 +-&//,9 01'%*1 β = +∞ ,'#(+1 +$' )*$;&;,-,%< 0&//,0& &--& +-&//1
)*1=&-1'%17
K13'1'#$ -& G(1/%,$'1 #& (' )('%$ #, =,/%& ;&L1/,&'$9 #&%& ('& #,/%*,2
;(>,$'1 & )*,$*, π(β, k) #, /())$*%$ [0, βmax] × {1, . . . , K}9 -& #,/%*,;(>,$'1
0&*.,'&-1 #, yn + 1 *,/(-%&
P(Yn+1 = Cj|"n+1, ,!) =
K∑
k=1
∫ βmax
0
P(Yn+1 = Cj|"n+1, ,!, β, k)π (β, k| ,!) dβ, ?@78B
#$=1 π (β, k| ,!) H -& #,/%*,;(>,$'1 & )$/%1*,$*, #&%$ -D,'/,101 ( ,!)7 M(12
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=1*$/,0,.-,&'>& f (y|x, β, k)N ,'5&%%,9 #&- 0$01'%$ +E1 -& +$/%&'%1 #, '$*0&-,>2
>&>,$'1 H #,)1'#1'%1 #& β 1 k9 '$' )$//,1#1 ('& 5$*0& 1/)-,+,%& #,*1%%&01'%1
+&-+$-&;,-17
O$01 &))*$//,0&>,$'1 /, (%,-,>>& -& )/1(#$2=1*$/,0,.-,&'>&
!"! #$%&'()*+$,- .-( &$.-(($ !
f̂ ( |!, β, k) =G∏
g=1
∏
yi=Cg
P(yi = Cg| −i,!, β, k),"#$%&
'()*+,+ -+. /)(-(,,( -0..0 /)(1+12.2,3 4(5-262(5+,0 -2 42+7485+ (770)9+62(50
-2 +//+),050)0 +..+ 78+ 90)+ 4.+770$ :0..+ "#$%& ; /)0705,0 85+ +//)(772*+<
62(50= -+. *(*05,( 4>0 .?25,0@)+62(50 -2 f̂ ( |", β, k) 5(5 9+.0 A 0 .+ 4(7,+5,0
-2 5()*+.266+62(50 5(5 ; 4(5(7428,+$ B2 /8C (992+)0 +. /)(1.0*+ 8,2.266+5-(
85 +.@()2,*( DEDE= 4>0 '()52740 85+ 7,2*+ -0..+ -27,)21862(50 + /(7,0)2()2
π̂ (β, k| ,!) ∝ f̂ ( |", β, k) π (β, k) 0 .+ /708-(<-27,)21862(50 /0) .0 /)09272(52
P̂(yn+1 = Cj|"n+1, ,!) =
K∑
k=1
∫ βmax
0
P(yn+1 = Cj|"n+1, ,!, β, k)π̂ (β, k| ,!) dβ. "#$#&
!"!" #$%% &'()*+*,-.$/0.(-12. 304+,')
F. 4+.4(.( 07/.242,( -0..+ "#$#& ; -2G42.*05,0 ,)+,,+12.0 0- ; )24>207,+ 85+
+//)(772*+62(50 DEDE= 1+7+,+ 78 85+ 4+,05+ -2 D+)H(9
(
(β(1), k(k),)
, . . . ,(
β(M), k(M)))
= 4(5 -27,)21862(50 7,+62(5+)2+ π̂ (β, k|y, x)= -+ 482 72*8.+)0 85
0.09+,( 58*0)( -2 9(.,0$
F5 85 /)2*( *(*05,(= /(,)0110 70*1)+)0 25,82,29( 8,2.266+)0 85( 74>0*+
-2 !""# #$%&'!()$ :(5 -27/(505-( /0)C -0..0 -27,)21862(52 4(5-262(5+,0
-2 β 0 k= ; )+@2(509(.0 /)(/())0 85 +.@()2,*( *+,-.&.'!#/0$#,!()# "72 90-+
+- 070*/2( I(10), 0 E+70..+= JKAK= E+/2,(.( &$ L?+@@2()5+*05,( -2 β 0
k +992050 8,2.266+5-( -27,)21862(52 /)(/(7,+ 4(5 /+770@@2+,+ 4+78+.0$ M+.
*(*05,( 4>0 β ∈ [0, βmax]= 72 /8C )24())0)0 + 85+ ,)+7'()*+62(50 .(@27,24+
-0. /+)+*0,)( θ= 25 482 β= 2590),05-( .+ )0.+62(50= +778*0 85+ 7,)8,,8)+ -0.
,2/(
β(t) = βmax exp(
θ(t))
/
(
exp(
θ(t))
+ 1)
, "#$N&
72*8.+5-( 2 9+.()2 -2 θ 4(5 /+770@@2+,+ 4+78+.0 -+ 1
(
θ(t), τ 2)
= 7056+ )07,)2<
62(52 +. 78//(),($ O0) .?+@@2()5+*05,( -2 k= 72 /8C 8,2.266+)0 85+ /)(/(7,+
852'()*0 78 r 924252 -2 k(t)= (990)( 78 85( 7/+62(
{
k(t) − r, . . . , k(t) + r}
∩{1, . . . , K}$ L+ -27,)21862(50 -0..0 /)(/(7,0 /0) k ; P825-2 Qr (k, ·) 4(5 '85<
62(50 -2 /)(1+12.2,3 qr(
k, k′)
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