Online Model Selection Based on the Variational Bayes

<strong>Online</strong> <strong>Model</strong> <strong>Selection</strong> <strong>Based</strong> on the Variational Bayes 1661
Z
C dm (µ )Qh (µ ) log ¡ P0(µ )/Qh (µ )¢ . (3.2)
The ratio (c 0/ T) determines the relative reliability between the observed
data and the prior belief for the parameter distribution. The expected free
energy, equation 3.2, can be estimated by
� ´
T Xt
Z
F(Xft g, Qzftg, Qh , T) D
t
tD1
dm (µ )Q Z
h (µ )
£ log ¡ P(x(t), z(t)| µ )/Qz(z(t)) ¢
Z
C
dm (z(t))Qz(z(t))
dm (µ )Q h (µ ) log ¡ P0(µ )/Q h (µ )¢ , (3.3)
where Qzftg D fQz(z(t))|t D 1, . . . , t g. Note that t represents the actual
amount of observed data, and it increases over time while T is �xed. The estimation
of the posterior distribution Qz(z(t)) is inaccurate in the early stage
of the online learning and gradually becomes accurate as learning proceeds.
However, the early inaccurate estimations and the later accurate estimations
contribute to the free energy (see equation 3.3) in equal weight. This might
cause slow convergence of the learning process. Therefore, we introduce a
time-dependent discount factor l(t) (0 · l(t) · 1, t D 2, 3, . . .) for forgetting
the earlier inaccurate estimation effects. Accordingly, a discounted free
energy is de�ned by
F l (Xft g, Qzft g, Q h , T) D Tg(t )
Z
£
tX
�tY tD1
l(s)
sDtC1
dm (µ )Qh (µ )
Z
´
dm (z(t))Qz(z(t))
£ log ¡ P(x(t), z(t)| µ )/Qz(z(t)) ¢
Z
C
where g(t ) represents a normalization constant:
g(t ) D
"
tX �tY tD1
l(s)
sDtC1
´#¡1
dm (µ )Q h (µ ) log ¡ P0(µ )/Q h (µ )¢ , (3.4)
. (3.5)
3.2 <strong>Online</strong> Variational Bayes Algorithm. The online VB algorithm can
be derived from the successive maximization of the discounted free energy
(see equation 3.4). Let us assume that Qzft ¡1g D fQz(z(t))|t D 1, . . . , t ¡1g
(µ ) have been determined for an observed data set Xft ¡ 1g D
and Q (t ¡1)
h