Online Model Selection Based on the Variational Bayes

1662 Masa-aki Sato
fx(t)|t D 1, . . . , t ¡ 1g. With new observed data x(t ), the discounted free
energy Fl (Xft g, Qzftg, Qh , T) is maximized with respect to Qz(z(t )), while
Qh is set to Q (t ¡1)
(µ ). The solution is given by
h
Qz(z(t )) D P(z(t )|x(t ), N µ (t ))
µ
N (t
Z
) D
dm (µ )Q(t ¡1)
(µ h
)µ. (3.6)
In the next step, the discounted free energy F l(Xft g, Qzftg, Q h , T) is maximized
with respect to Q h (µ ), while Qzft g is �xed. The solution is given
by
Q (t )
h (µ ) D exp [c (® (t ) ¢ µ ¡ Ã (µ )) ¡ © (® (t ), c )] ,
c D T C c 0,
c ® (t ) D Thhr(x, z)ii(t ) C c 0®0, (3.7)
where the discounted average hh¢ii(t ) is de�ned by
hhr(x, z)ii(t ) D g(t )
Z
£
tX
�tY tD1
l(s)
sDtC1
´
dm (z(t))Qz(z(t))r(x(t), z(t)). (3.8)
The discounted average can be calculated by using a step-wise equation:
hhr(x, z)ii(t ) D (1 ¡ g(t ))hhr(x, z)ii(t ¡ 1)
h
C g(t )Ez r(x(t ), z(t ))| µ
N (t ) i
,
g(t ) D (1 C l(t )/g(t ¡ 1)) ¡1 . (3.9)
By using equation 3.6, the expectation value of the suf�cient statistics for
the current data Ez[r(x(t ), z(t ))| N µ (t )] is given by
Ez
h
r(x(t ), z(t ))| µ
N (t ) i Z
D dm (z(t ))
£ P(z(t )|x(t ), N µ (t ))r(x(t ), z(t )). (3.10)
The recursive formula for ® (t ) is derived from the above equations:
D® (t ) D ® (t ) ¡ ® (t ¡ 1)