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Fig. 10.1
Laplacian distribution with
μ
=
0and
b
=
1
M-Step
: There is only one parameter in Eq. (
10.4
) i.e.,
b
m
.Nowwehavethe
model parameter set given by
=[
ʱ
,
]
ʸ
b
m
for each mixture component. Using
m
m
M
m
=
1
ʱ
=
the maximum likelihood principle, and enforcing the condition
1, we
m
get the following update equations for
b
m
and
ʱ
m
;
z
(
i
m
x
(
i
)
N
∑
i
=
1
b
m
=
z
(
i
m
,
(10.6)
N
∑
i
=
1
z
(
i
m
N
∑
i
=
1
=
ʱ
(10.7)
m
N
where
·
is the expectation operator.
x
(
1
)
,...,
x
(
N
)
|
ʘ
)
E-Step
:
To
maximize
the
incomplete
log-likelihood
p
(
,
we
take
the
expectation
w.r.t.
the
posterior
distribution
of
Z
namely
z
(
i
)
N
i
=
1
x
(
1
)
,...,
x
(
N
)
,
ʘ
)
(
|
=
. Therefore, each of the expectations
z
(
i
m
that appear in the above update equations is computed as follows:
p
Z
, where
Z
z
(
i
m
x
(
i
)
|
ʸ
p
(
)
ʱ
m
m
=
(10.8)
M
j
=
1
p
x
(
i
)
|
ʸ
j
)
ʱ
j
(
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