Online Model Selection Based on the Variational Bayes
Online Model Selection Based on the Variational Bayes
Online Model Selection Based on the Variational Bayes
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<str<strong>on</strong>g>Online</str<strong>on</strong>g> <str<strong>on</strong>g>Model</str<strong>on</strong>g> <str<strong>on</strong>g>Selecti<strong>on</strong></str<strong>on</strong>g> <str<strong>on</strong>g>Based</str<strong>on</strong>g> <strong>on</strong> <strong>the</strong> Variati<strong>on</strong>al <strong>Bayes</strong> 1675<br />
C Qh (µ )-independent terms<br />
Z<br />
D dm (ZfTg)Qz(ZfTg)<br />
" TX<br />
#<br />
£ (r(x(t), Z(t)) ¢ hµiQ C r0(x(t), Z(t))) ¡ log Qz(ZfTg)<br />
h<br />
tD1<br />
C Qz(ZfTg) - independent terms,<br />
where h¢iQ and h¢iQz denote <strong>the</strong> expectati<strong>on</strong> value with respect to Q h h (µ ) and<br />
Qz(ZfTg), respectively.<br />
Appendix B<br />
The calculati<strong>on</strong> of <strong>the</strong> derivative of <strong>the</strong> parameterized free energy (see equati<strong>on</strong><br />
2.26) is lengthy but straightforward. The outline of <strong>the</strong> calculati<strong>on</strong> is<br />
shown below. The derivative with respect to µ<br />
N can be calculated as<br />
@F/@ N �<br />
@<br />
µ D T<br />
@N µ hr(x, z)i ´<br />
N (hµi<br />
µ ® ¡ µ N ),<br />
by using <strong>the</strong> relati<strong>on</strong><br />
@<br />
@N �TX ´ �<br />
log P(x(t)| µ<br />
N ) D T hr(x, z)i N ¡<br />
µ<br />
µ @Ã<br />
@N ´<br />
.<br />
µ<br />
tD1<br />
The coef�cient matrix T(@hri N /@<br />
µ N µ ) turns out to be U(µ ) de�ned in equati<strong>on</strong><br />
2.28.<br />
The derivatives with respect to (®, c ) are given by<br />
1<br />
c<br />
@F<br />
@® D<br />
�<br />
1<br />
c 2<br />
@ 2 W<br />
@®@® T<br />
´<br />
¢ ± ²<br />
Thr(x, z)i N C c 0®0 ¡ (T C c 0)®<br />
µ<br />
�<br />
1 @<br />
C (T C c 0 ¡c )<br />
c<br />
2 W 1<br />
¡<br />
@®@c c 2<br />
´<br />
@W<br />
,<br />
@®<br />
1<br />
¡<br />
c 2<br />
´<br />
@W<br />
¢<br />
@®<br />
± ²<br />
Thr(x, z)i N C c 0®0 ¡ (T C c 0)®<br />
µ<br />
� 2 ´<br />
@ W<br />
C (T C c 0 ¡c ) .<br />
@c @c<br />
@F<br />
@c D<br />
�<br />
1 @<br />
c<br />
2 W<br />
@®@c<br />
Equati<strong>on</strong>s 2.30 and 2.31 can be derived by using <strong>the</strong> above and <strong>the</strong> following<br />
equati<strong>on</strong>s:<br />
1<br />
c 2<br />
@ 2 W 1<br />
D<br />
T @®@® c 2<br />
½ � @ log P a<br />
@®<br />
´ �<br />
@ log Pa @® T<br />
´¾<br />
,<br />
®
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