Biomedical Engineering Reference
In-Depth Information
we have
ʲ
ʲ
∂
F
∂
ʴ
I
=
ʴ
F
(
x
)
dx
=
f
ʴ
f
(
x
)
dx
,
(C.75)
ʱ
ʱ
and thus,
ʴ
I
f
=
∂
F
∂
f
.
(C.76)
ʴ
As an example of the use of Eq. (
C.76
), let us derive the derivative of the functional
L[
q
(
x
)
]
in Eq. (
B.53
). Ignoring the last term, which does not contain
q
(
x
)
,Eq.(
B.53
)
is rewritten as
∞
L[
q
(
x
)
]=
d
x
q
(
x
)
log
p
(
x
,
y
|
ʸ
)
−∞
∞
∞
−
d
x
q
(
x
)
log
q
(
x
)
+
ʳ
q
(
x
)
d
x
.
(C.77)
−∞
−∞
In the first term, since
F
(
x
)
=
q
(
x
)
log
p
(
x
,
y
|
ʸ
)
,wehave
∂
F
q
=
log
p
(
x
,
y
|
ʸ
).
(C.78)
∂
In the second term, since
F
(
x
)
=
q
(
x
)
log
q
(
x
)
,wehave
q
)
=
∂
F
1
1
q
=
(
x
)
log
q
(
x
log
q
(
x
)
+
q
(
x
)
)
=
log
q
(
x
)
+
1
.
(C.79)
∂
∂
q
(
x
)
q
(
x
In the third term, since
F
(
x
)
=
ʳ
q
(
x
)
,wehave
∂
F
∂
q
=
ʳ.
(C.80)
Therefore, we have
∂
L[
q
(
x
)
]
=
log
p
(
x
,
y
|
ʸ
)
−
log
q
(
x
)
−
1
+
ʳ,
(C.81)
∂
q
(
x
)
which is equal to Eq. (
B.54
).
C.6
Vector and Matrix Derivatives
Differentiating a scalar
F
with a column vector
x
is defined as creating a column
vector whose
j
th element is equal to
∂
/∂
x
j
.
Assuming that
a
is a column vector and
A
is a matrix, The following relationships hold,
F
