The Azimuth Project
Separable function
Separable function
Idea
Details
Idea
Details
Multiplicatively separable
(1)
F
(
Θ
,
x
)
=
∏
i
=
1
:
n
f
i
(
θ
i
,
x
)
F(\Theta,\mathbf{x})=\prod_{i=1:n} f_i(\theta_i,\mathbf{x})
Then
(2)
∂
F
(
Θ
,
x
)
∂
θ
j
=
∂
f
j
(
θ
j
,
x
)
∂
θ
j
∏
i
=
1
:
n
,
i
≠
j
f
i
(
θ
i
,
x
)
\frac{\partial F(\Theta,\mathbf{x})}{\partial \theta_j} =\frac{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j} \prod_{i=1:n,i \ne j} f_i(\theta_i,\mathbf{x})
(3)
∂
2
F
(
Θ
,
x
)
∂
θ
j
2
=
∂
2
f
j
(
θ
j
,
x
)
∂
θ
j
2
∏
i
=
1
:
n
,
i
≠
j
f
i
(
θ
i
,
x
)
\frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j^2} =\frac{\partial^2 f_j(\theta_j,\mathbf{x})}{\partial \theta_j^2} \prod_{i=1:n,i \ne j} f_i(\theta_i,\mathbf{x})
(4)
∂
2
F
(
Θ
,
x
)
∂
θ
j
∂
θ
k
=
∂
f
j
(
θ
j
,
x
)
∂
θ
j
∂
f
k
(
θ
k
,
x
)
∂
θ
k
∏
i
=
1
:
n
,
i
≠
j
,
k
f
i
(
θ
i
,
x
)
\frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j \partial \theta_k} =\frac{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j} \frac{\partial f_k(\theta_k,\mathbf{x})}{\partial \theta_k} \prod_{i=1:n,i \ne j,k} f_i(\theta_i,\mathbf{x})
Special case:
(5)
F
(
Θ
,
x
)
=
exp
∑
i
=
1
:
n
f
i
(
θ
i
,
x
)
F(\Theta,\mathbf{x})=\exp \sum_{i=1:n} f_i(\theta_i,\mathbf{x})
Then
(6)
∂
F
(
Θ
,
x
)
∂
θ
j
=
exp
(
∂
f
j
(
θ
j
,
x
)
∂
θ
j
+
∑
i
=
1
:
n
,
i
≠
j
f
i
(
θ
i
,
x
)
)
\frac{\partial F(\Theta,\mathbf{x})}{\partial \theta_j} =\exp (\frac{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j} +\sum_{i=1:n,i \ne j} f_i(\theta_i,\mathbf{x}))
(7)
∂
2
F
(
Θ
,
x
)
∂
θ
j
2
=
exp
(
∂
2
f
j
(
θ
j
,
x
)
∂
θ
j
2
+
∑
i
=
1
:
n
,
i
≠
j
f
i
(
θ
i
,
x
)
)
\frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j^2} =\exp (\frac{\partial^2 f_j(\theta_j,\mathbf{x})}{\partial \theta_j^2} +\sum_{i=1:n,i \ne j} f_i(\theta_i,\mathbf{x}) )
(8)
∂
2
F
(
Θ
,
x
)
∂
θ
j
∂
θ
k
=
exp
(
∂
f
j
(
θ
j
,
x
)
∂
θ
j
+
∂
f
k
(
θ
k
,
x
)
∂
θ
k
+
∑
i
=
1
:
n
,
i
≠
j
,
k
f
i
(
θ
i
,
x
)
)
\frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j \partial \theta_k} =\exp (\frac{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j} +\frac{\partial f_k(\theta_k,\mathbf{x})}{\partial \theta_k} +\sum_{i=1:n,i \ne j,k} f_i(\theta_i,\mathbf{x}) )
category:
mathematical methods
Revised on March 20, 2011 02:40:47 by
David Tweed