两个复高斯分布的乘积(高维) |
您所在的位置:网站首页 › 高斯分布 相乘 › 两个复高斯分布的乘积(高维) |
假设 p ( x ) ∝ C N ( μ 1 , Σ 1 ) C N ( μ 2 , Σ 2 ) p(\boldsymbol x) \propto \mathcal{CN}(\boldsymbol \mu_1, \boldsymbol \Sigma_1)\mathcal{CN}(\boldsymbol \mu_2, \boldsymbol \Sigma_2) p(x)∝CN(μ1,Σ1)CN(μ2,Σ2),有 p ( x ) ∝ exp { − ( x − μ 1 ) H Σ 1 − 1 ( x − μ 1 ) − ( x − μ 2 ) H Σ 2 − 1 ( x − μ 2 ) } ∝ exp { − x H Σ 1 − 1 x + 2 R { μ 1 H Σ 1 − 1 x } − x H Σ 2 − 1 x + 2 R { μ 2 H Σ 2 − 1 x } } = exp { − x H ( Σ 1 − 1 + Σ 2 − 1 ) x + 2 R { ( μ 1 H Σ 1 − 1 + μ 2 H Σ 2 − 1 ) x } } = exp { − [ x H [ Σ 1 − 1 + Σ 2 − 1 ⏟ = Σ x − 1 ] x − 2 R { [ Σ x ( Σ 1 − 1 μ 1 + Σ 2 − 1 μ 2 ) ⏟ = μ x ] H Σ x − 1 x } ] } = exp { − [ x H Σ x − 1 x + 2 R μ x H Σ x − 1 x ] } ∝ exp { − ( x − μ x ) H Σ − 1 ( x − μ x ) } \begin{aligned} p(\boldsymbol x)&\propto \exp{\left \{ -{(\boldsymbol x - \boldsymbol \mu_1)^H \boldsymbol \Sigma^{-1}_1 (\boldsymbol x - \boldsymbol \mu_1)}{} -{(\boldsymbol x - \boldsymbol \mu_2)^H \boldsymbol \Sigma^{-1}_2 (\boldsymbol x - \boldsymbol \mu_2)}{} \right \}} \\ &\propto \exp{\left \{ -{\boldsymbol x^H \boldsymbol \Sigma^{-1}_1 \boldsymbol x + 2 \mathcal{R} \left \{\boldsymbol \mu^H_1 \boldsymbol \Sigma^{-1}_1 \boldsymbol x\right \}}{} - {\boldsymbol x^H \boldsymbol \Sigma^{-1}_2 \boldsymbol x + 2 \mathcal{R} \left \{\boldsymbol \mu^H_2 \boldsymbol \Sigma^{-1}_2 \boldsymbol x\right \}}{} \right \}} \\ &=\exp{\left \{ {-\boldsymbol x^H \left ( \boldsymbol \Sigma^{-1}_1 + \boldsymbol \Sigma^{-1}_2 \right ) \boldsymbol x + 2 \mathcal{R} \left \{ \left( \boldsymbol \mu^H_1 \boldsymbol \Sigma^{-1}_1 + \boldsymbol \mu^H_2 \boldsymbol \Sigma^{-1}_2 \right ) \boldsymbol x \right \}}{} \right \}} \\ & = \exp \left \{ - \left [\boldsymbol x^H \left [ \mathop {\underbrace{{\boldsymbol{\varSigma }_{1}^{-1}+\boldsymbol{\varSigma }_{2}^{-1}}{}}} \limits_{=\boldsymbol{\varSigma }_{x}^{-1}} \right ] \boldsymbol x - 2 \mathcal{ R } \left \{ \left[ \mathop {\underbrace{{\boldsymbol{\varSigma }_x\left( \boldsymbol{\varSigma }_{1}^{-1}\boldsymbol{\mu }_1+\boldsymbol{\varSigma }_{2}^{-1}\boldsymbol{\mu }_2 \right)}{}}} \limits_{=\boldsymbol{\mu }_x} \right] ^H \boldsymbol \Sigma^{-1}_x \boldsymbol x \right \} \right ]\right \} \\ &= \exp \left \{ - \left [\boldsymbol x^H \boldsymbol \Sigma^{-1}_x \boldsymbol x + 2 \mathcal{ R } \boldsymbol \mu^H_x \boldsymbol \Sigma^{-1}_x \boldsymbol x \right ] \right \} \\ & \propto \exp \left \{ - (\boldsymbol x - \boldsymbol \mu_x)^H \boldsymbol \Sigma^{-1} (\boldsymbol x - \boldsymbol \mu_x) \right \} \end{aligned} p(x)∝exp{−(x−μ1)HΣ1−1(x−μ1)−(x−μ2)HΣ2−1(x−μ2)}∝exp{−xHΣ1−1x+2R{μ1HΣ1−1x}−xHΣ2−1x+2R{μ2HΣ2−1x}}=exp{−xH(Σ1−1+Σ2−1)x+2R{(μ1HΣ1−1+μ2HΣ2−1)x}}=exp⎩⎪⎪⎨⎪⎪⎧−⎣⎢⎢⎡xH⎣⎢⎡=Σx−1 Σ1−1+Σ2−1⎦⎥⎤x−2R⎩⎪⎪⎨⎪⎪⎧⎣⎢⎡=μx Σx(Σ1−1μ1+Σ2−1μ2)⎦⎥⎤HΣx−1x⎭⎪⎪⎬⎪⎪⎫⎦⎥⎥⎤⎭⎪⎪⎬⎪⎪⎫=exp{−[xHΣx−1x+2RμxHΣx−1x]}∝exp{−(x−μx)HΣ−1(x−μx)} 因此 x ∼ C N ( μ x , Σ x ) x \sim \mathcal {CN}(\boldsymbol \mu_x, \boldsymbol \Sigma_x) x∼CN(μx,Σx),其中 μ x = Σ x ( Σ 1 − 1 μ 1 + Σ 2 − 1 μ 2 ) Σ x = ( Σ 1 − 1 + Σ 2 − 1 ) − 1 \begin{aligned} \boldsymbol \mu_x & = \boldsymbol{\varSigma }_x\left( \boldsymbol{\varSigma }_{1}^{-1}\boldsymbol{\mu }_1+\boldsymbol{\varSigma }_{2}^{-1}\boldsymbol{\mu }_2 \right) \\ \boldsymbol \Sigma_x &= { \left (\boldsymbol \Sigma^{-1}_1 + \boldsymbol \Sigma^{-1}_2 \right )}^{-1} \end{aligned} μxΣx=Σx(Σ1−1μ1+Σ2−1μ2)=(Σ1−1+Σ2−1)−1 注意,当 x \boldsymbol x x为标量 x x x时, x ∼ C N ( μ x , σ x ) x \sim \mathcal{CN}(\mu_x, \sigma_x) x∼CN(μx,σx) σ x = ( 1 σ 1 2 + 1 σ 2 2 ) − 1 = σ 1 2 σ 2 2 σ 1 2 + σ 2 2 μ x = σ 1 2 σ 2 2 σ 1 2 + σ 2 2 ( μ 1 σ 1 2 + μ 2 σ 2 2 ) \begin{aligned} \sigma_x &= {\left ( \frac{1}{\sigma^2_1}+\frac{1}{\sigma^2_2} \right )}^{-1}= \frac{\sigma^2_1 \sigma^2_2}{\sigma^2_1+\sigma^2_2}\\ \mu_x &= \frac{\sigma^2_1 \sigma^2_2}{\sigma^2_1+\sigma^2_2} \left ( \frac{\mu_1}{\sigma^2_1} + \frac{\mu_2}{\sigma^2_2} \right ) \end{aligned} σxμx=(σ121+σ221)−1=σ12+σ22σ12σ22=σ12+σ22σ12σ22(σ12μ1+σ22μ2) 然而,在实际编写代码的过程中,考虑到数值的稳定性,我们一般按照如下顺序执行: g = σ 1 2 σ 1 2 + σ 2 2 μ x = g ⋅ ( μ 2 − μ 1 ) + μ 1 σ x = g ⋅ σ 1 2 \begin{aligned} g & = \frac{\sigma^2_1 }{\sigma^2_1+\sigma^2_2} \\ \mu_x &= g \cdot (\mu_2 - \mu_1) + \mu_1 \\ \sigma_x &= g \cdot \sigma^2_1 \end{aligned} gμxσx=σ12+σ22σ12=g⋅(μ2−μ1)+μ1=g⋅σ12 另外,考虑 常见的线性模型 \textbf{常见的线性模型} 常见的线性模型: y = A x + w \boldsymbol y = \boldsymbol {Ax} + \boldsymbol w y=Ax+w 其中 x \boldsymbol x x的先验分布: x ∼ C N ( x ; r , Σ 1 ) \boldsymbol x \sim \mathcal {CN}(\boldsymbol x; \boldsymbol r, \boldsymbol \Sigma_1) x∼CN(x;r,Σ1),似然分布 y ∣ A x ∼ C N ( y ; A x , Σ 2 ) \boldsymbol y | \boldsymbol {Ax} \sim \mathcal{CN}(\boldsymbol y; \boldsymbol{Ax}, \boldsymbol \Sigma_2) y∣Ax∼CN(y;Ax,Σ2),则关于 x \boldsymbol x x的后验分布: p ( x ∣ y ) ∝ C N ( x ; r , Σ 1 ) ⋅ C N ( y ; A x , Σ 2 ) ∝ exp { − ( x − r ) H Σ 1 − 1 ( x − r ) − ( A x − y ) H Σ 2 − 1 ( A x − y ) } ∝ exp { x H Σ 1 − 1 x − 2 R { r H Σ 1 − 1 x } − x H A H Σ 2 − 1 A x − 2 R { y H Σ 2 − 1 A x } } = exp { x H ( Σ 1 − 1 + A H Σ 2 − 1 A ) x − 2 R { ( r H Σ 1 − 1 + y H Σ 2 − 1 A ) x } } = exp { − x H [ Σ 1 − 1 + A H Σ 2 − 1 A ⏟ = Σ x − 1 ] x + 2 R { [ Σ x ( Σ 1 − 1 r + A H Σ 2 − 1 y ) ⏟ = μ x ] H Σ x − 1 x } } \begin{aligned} p(\boldsymbol x| \boldsymbol y) &\propto \mathcal {CN}(\boldsymbol x; \boldsymbol r, \boldsymbol \Sigma_1) \cdot \mathcal{CN}(\boldsymbol y; \boldsymbol{Ax}, \boldsymbol \Sigma_2) \\ & \propto \exp{\left \{ -{(\boldsymbol x - \boldsymbol r)^H \boldsymbol \Sigma^{-1}_1 (\boldsymbol x - \boldsymbol r)}{} -{(\boldsymbol {Ax} - \boldsymbol y)^H \boldsymbol \Sigma^{-1}_2 (\boldsymbol {Ax} - \boldsymbol y)}{} \right \}} \\ & \propto \exp{\left \{ {\boldsymbol x^H \boldsymbol \Sigma^{-1}_1 \boldsymbol x - 2 \mathcal{R} \left \{\boldsymbol r^H \boldsymbol \Sigma^{-1}_1 \boldsymbol x\right \}}{} - {\boldsymbol x^H \boldsymbol A^H \boldsymbol \Sigma^{-1}_2 \boldsymbol A \boldsymbol x - 2 \mathcal{R} \left \{\boldsymbol y^H \boldsymbol \Sigma^{-1}_2 \boldsymbol A \boldsymbol x\right \}}{} \right \}} \\ &= \exp{\left \{ {\boldsymbol x^H \left ( \boldsymbol \Sigma^{-1}_1 + \boldsymbol A^H \boldsymbol \Sigma^{-1}_2 \boldsymbol A \right ) \boldsymbol x - 2 \mathcal{R} \left \{ \left( \boldsymbol r^H \boldsymbol \Sigma^{-1}_1 + \boldsymbol y^H \boldsymbol \Sigma^{-1}_2 \boldsymbol A \right ) \boldsymbol x \right \}}{} \right \}} \\ & = \exp \left \{ -\boldsymbol x^H \left [ \mathop {\underbrace{{\boldsymbol{\varSigma }_{1}^{-1}+{\boldsymbol A^H \boldsymbol \Sigma^{-1}_2 \boldsymbol A}^{}}{}}} \limits_{=\boldsymbol{\varSigma }_{x}^{-1}} \right ] \boldsymbol x + 2 \mathcal{ R } \left \{ \left[ \mathop {\underbrace{{\boldsymbol{\varSigma }_x\left( \boldsymbol{\varSigma }_{1}^{-1}\boldsymbol r+\boldsymbol A^H \boldsymbol{\varSigma }_{2}^{-1}\boldsymbol{y } \right)}{}}} \limits_{=\boldsymbol{\mu }_x} \right] ^H \boldsymbol \Sigma^{-1}_x \boldsymbol x \right \} \right \} \\ \end{aligned} p(x∣y)∝CN(x;r,Σ1)⋅CN(y;Ax,Σ2)∝exp{−(x−r)HΣ1−1(x−r)−(Ax−y)HΣ2−1(Ax−y)}∝exp{xHΣ1−1x−2R{rHΣ1−1x}−xHAHΣ2−1Ax−2R{yHΣ2−1Ax}}=exp{xH(Σ1−1+AHΣ2−1A)x−2R{(rHΣ1−1+yHΣ2−1A)x}}=exp⎩⎪⎪⎨⎪⎪⎧−xH⎣⎢⎡=Σx−1 Σ1−1+AHΣ2−1A⎦⎥⎤x+2R⎩⎪⎪⎨⎪⎪⎧⎣⎢⎢⎡=μx Σx(Σ1−1r+AHΣ2−1y)⎦⎥⎥⎤HΣx−1x⎭⎪⎪⎬⎪⎪⎫⎭⎪⎪⎬⎪⎪⎫ 所以有 Σ x = ( Σ 1 − 1 + A H Σ 2 − 1 A ) − 1 x = ( Σ 1 − 1 + A H Σ 2 − 1 A ) − 1 ( Σ 1 − 1 r + A H Σ 2 − 1 y ) \begin{aligned} \boldsymbol \Sigma_x &= {\left ( \boldsymbol{\varSigma }_{1}^{-1}+\boldsymbol A^H \boldsymbol \Sigma^{-1}_2 \boldsymbol A ^{} \right )}^{-1} \\ \boldsymbol x &={\left ( \boldsymbol{\varSigma }_{1}^{-1}+\boldsymbol A^H \boldsymbol \Sigma^{-1}_2 \boldsymbol A ^{} \right )}^{-1} \left( \boldsymbol{\varSigma }_{1}^{-1}\boldsymbol r+\boldsymbol A^H \boldsymbol{\varSigma }_{2}^{-1}\boldsymbol{y } \right) \end{aligned} Σxx=(Σ1−1+AHΣ2−1A)−1=(Σ1−1+AHΣ2−1A)−1(Σ1−1r+AHΣ2−1y) 事实上, x = E [ x ∣ y ] \boldsymbol x = \mathbb E[\boldsymbol x| \boldsymbol y] x=E[x∣y],所以该均值也是MMSE估计的结果,因为其估计结果是线性的,所以也称为“LMMSE”。 |
CopyRight 2018-2019 办公设备维修网 版权所有 豫ICP备15022753号-3 |