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高斯PDF
标量实高斯分布 N ( x ∣ a , A ) = 1 2 π A exp [ − ( x − a ) 2 2 A ] \mathcal{N}(x|a,A)=\frac{1}{\sqrt{2\pi A} }\exp \left[{-\frac{(x-a)^2}{2A} }\right] N(x∣a,A)=2πA
1exp[−2A(x−a)2]
标量复高斯分布 N c ( x ∣ a , A ) = 1 π A exp [ − ∣ ∣ x − a ∣ ∣ 2 A ] \mathcal{N}_c(x|a,A)=\frac{1}{\pi A}\exp \left[-\frac{||x-a||^2}{A}\right] Nc(x∣a,A)=πA1exp[−A∣∣x−a∣∣2]
矢量实高斯分布 N ( x ∣ a , A ) = ( 2 π ) − N 2 det ( A ) − 1 2 exp ( − 1 2 ( x − a ) T A − 1 ( x − a ) ) \mathcal{N}(\boldsymbol{x}|\boldsymbol{a},\boldsymbol{A})=(2\pi)^{-\frac{N}{2} }\det(\boldsymbol{A})^{-\frac{1}{2} }\exp \left({-\frac{1}{2}(\boldsymbol{x}-\boldsymbol{a})^T\boldsymbol{A}^{-1}(\boldsymbol{x}-\boldsymbol{a})}\right) N(x∣a,A)=(2π)−2Ndet(A)−21exp(−21(x−a)TA−1(x−a)) 其中 N N N表示 x \boldsymbol{x} x的维度。
矢量复高斯分布 N c ( x ∣ a , A ) = 1 det ( π A ) exp [ − ( x − a ) H A − 1 ( x − a ) ] \mathcal{N}_c(\boldsymbol{x}|\boldsymbol{a},\boldsymbol{A})=\frac{1}{\det(\pi \boldsymbol{A})}\exp \left[{-(\boldsymbol{x}-\boldsymbol{a})^H\boldsymbol{A}^{-1}(\boldsymbol{x}-\boldsymbol{a})}\right] Nc(x∣a,A)=det(πA)1exp[−(x−a)HA−1(x−a)]
标量实高斯相乘引理
给定高斯概率分布 N ( x ∣ a , A ) \mathcal{N}(x|a,A) N(x∣a,A)和 N ( x ∣ b , B ) \mathcal{N}(x|b,B) N(x∣b,B),存在 N ( x ∣ a , A ) N ( x ∣ b , B ) = N ( 0 ∣ a − b , A + B ) N ( x ∣ a A + b B 1 A + 1 B , 1 1 A + 1 B ) \mathcal{N}(x|a,A)\mathcal{N}(x|b,B)=\mathcal{N}(0|a-b,A+B)\mathcal{N} \left({x\left|\frac{\frac{a}{A}+\frac{b}{B} }{\frac{1}{A}+\frac{1}{B} },\frac{1}{\frac{1}{A}+\frac{1}{B} }\right.}\right) N(x∣a,A)N(x∣b,B)=N(0∣a−b,A+B)N(x∣∣∣∣∣A1+
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