设
X
X
X
为服从多维正态分布的随机变量,即
X
∼
N
(
μ
,
Σ
)
X\sim N(\mu,\Sigma)
X
∼
N
(
μ
,
Σ
)
将
X
X
X
分成两个部分:
X
a
,
X
b
X_a,X_b
X
a
,
X
b
,
μ
\mu
μ
分成两个部分:
μ
a
,
μ
b
\mu_a, \mu_b
μ
a
,
μ
b
,
Σ
\Sigma
Σ
分成四个部分:
Σ
a
a
,
Σ
a
b
,
Σ
b
a
,
Σ
b
b
\Sigma_{aa},\Sigma_{ab},\Sigma_{ba},\Sigma_{bb}
Σ
a
a
,
Σ
a
b
,
Σ
b
a
,
Σ
b
b
目标:求解
P
(
X
a
)
P(X_a)
P
(
X
a
)
与
P
(
X
b
∣
X
a
)
P(X_b|X_a)
P
(
X
b
∣
X
a
)
定理:
已知
X
∼
N
(
μ
,
Σ
)
X\sim N(\mu, \Sigma)
X
∼
N
(
μ
,
Σ
)
,
Y
=
A
X
+
B
Y=AX+B
Y
=
A
X
+
B
则
Y
∼
N
(
A
μ
+
B
,
A
Σ
A
T
)
Y\sim N(A\mu +B, A\Sigma A^T)
Y
∼
N
(
A
μ
+
B
,
A
Σ
A
T
)
结论:
X
a
∼
N
(
μ
a
,
Σ
a
a
)
X_a\sim N(\mu_a,\Sigma_{aa})
X
a
∼
N
(
μ
a
,
Σ
a
a
)
令:
X
b
⋅
a
=
X
b
−
Σ
b
a
Σ
a
a
−
1
X
a
X_{b\cdot a}=X_b-\Sigma_{ba}\Sigma_{aa}^{-1}X_a
X
b
⋅
a
=
X
b
−
Σ
b
a
Σ
a
a
−
1
X
a
μ
b
⋅
a
=
μ
b
−
Σ
b
a
Σ
a
a
−
1
μ
a
\mu_{b\cdot a}=\mu_b-\Sigma_{ba}\Sigma_{aa}^{-1}\mu_a
μ
b
⋅
a
=
μ
b
−
Σ
b
a
Σ
a
a
−
1
μ
a
Σ
b
b
⋅
a
=
Σ
b
b
−
Σ
b
a
Σ
a
a
−
1
Σ
a
b
\Sigma_{bb\cdot a}=\Sigma_{bb}-\Sigma_{ba}\Sigma_{aa}^{-1}\Sigma_{ab}
Σ
b
b
⋅
a
=
Σ
b
b
−
Σ
b
a
Σ
a
a
−
1
Σ
a
b
结论:
X
b
∣
X
a
∼
N
(
μ
b
⋅
a
+
Σ
b
a
Σ
a
a
−
1
X
a
,
Σ
b
b
⋅
a
)
X_b|X_a\sim N(\mu_{b\cdot a}+\Sigma_{ba}\Sigma_{aa}^{-1}X_a,\Sigma_{bb\cdot a})
X
b
∣
X
a
∼
N
(
μ
b
⋅
a
+
Σ
b
a
Σ
a
a
−
1
X
a
,
Σ
b
b
⋅
a
)