X1
X2
.
Xn
Y1
Y2
.
Yn
Z1
Z2
.
Zn
Figure 1: The sigmoid network
2
BP in Sigmoid Networks
Solution due to Steve Gardiner
1. Write down the belief propagation updates for the network.
Figure 2 shows the factor graph of the network. Messages from Xi to the X,
And X(ej2f ) = Xc (f /T ), given below
X(ej2f )
61
1/2
1
20
1
20
1
6
f
(3) x[n] = xc (n/30) = cos(n/3) + sin(n/10 + /2) = cos(n/3) + cos(n/10).
(4) Here the sampling frequency
fs = 1/T = 8Hz. The FT of xc (t) is the same. The FT
P
of xs (t) is Xs (f ) =
D) its reserves must equal 50% of its money supply.
54. If domestic credit is constant, then any change in the demand for money will result in:
A) a change in foreign credit.
B) a change in the rate of interest.
C) an inverse change in the price level.
D)
P
k)
= n nk xn . Next, we maximize each clusters regression coefficient vector k . We can
because J(
k
analytically solve for k by setting its derivative to zero.
k)
J(
k
=
X
nk
n
0 =
X
cfw_(yn kT xn )2 = 0
k
nk (yn k xn )xn
(23)
(24)
n
Then solving t
HMM result See Figure 1a for log-likelihood. The log-likelihood score on the test data (2.5517 103)
was approximately similar to that of the train data (2.4726 103 ). The results of HMM training and
test are visualized in Figure 1b and 1c, respectively. N
3.2
Marginals
By Bayes rule,
P (cfw_xi ) =
by Eq.2, 3
by Eq.4
P (cfw_hj , xi )
P (cfw_hj |cfw_xi )
X
X
X
X
jb
ia fia (xi ) +
jb gjb (hj ) +
Wia
fia (xi )gjb (hj )
jb gjb (hj )
exp
i,a
j,b
i,j,a,b
jb
X
ia fia (xi )
exp
i,a
Then applying Eq.9, we essenti