两个高斯分布相乘：
N ( x ; μ 1 , ν 1 ) N ( x ; μ 2 , ν 2 ) = 1 2 π ν 1 exp ( − ( x − μ 1 ) 2 2 ν 1 ) ⋅ 1 2 π ν 2 exp ( − ( x − μ 2 ) 2 2 ν 2 ) = 1 2 π ν 1 ν 2 exp { − ν 2 ( x 2 − 2 μ 1 x + μ 1 2 ) 2 ν 1 ν 2 − ν 1 ( x 2 − 2 μ 2 x + μ 2 2 ) 2 ν 1 ν 2 } = 1 2 π ν 1 ν 2 exp { − ( ν 1 + ν 2 ) x 2 − 2 ( μ 1 ν 2 + μ 2 ν 1 ) x + μ 1 2 ν 2 + μ 2 2 ν 1 2 ν 1 ν 2 } = 1 2 π ν 1 ν 2 exp { − x 2 − 2 ( μ 1 ν 2 + μ 2 ν 1 ) ν 1 + ν 2 x + ( ( μ 1 ν 2 + μ 2 ν 1 ) ν 1 + ν 2 ) 2 + μ 1 2 ν 2 + μ 2 2 ν 1 ν 1 + ν 2 − ( ( μ 1 ν 2 + μ 2 ν 1 ) ν 1 + ν 2 ) 2 2 ν 1 ν 2 ν 1 + ν 2 } = 1 2 π ( ν 1 + ν 2 ) exp { − μ 1 2 ν 2 + μ 2 2 ν 1 − ( μ 1 ν 2 + μ 2 ν 1 ) 2 ν 1 + ν 2 2 ν 1 ν 2 } ⋅ 1 2 π ν 1 ν 2 ν 1 + ν 2 exp { − ( x − μ 1 ν 2 + μ 2 ν 1 ν 1 + ν 2 ) 2 2 ν 1 ν 2 ν 1 + ν 2 } = 1 2 π ( ν 1 + ν 2 ) exp { − ( μ 1 − μ 2 ) 2 2 ( ν 1 + ν 2 ) } ⋅ 1 2 π ν 1 ν 2 ν 1 + ν 2 exp { − ( x − μ 1 ν 2 + μ 2 ν 1 ν 1 + ν 2 ) 2 2 ν 1 ν 2 ν 1 + ν 2 } = N ( μ 1 ; μ 2 , ( ν 1 + ν 2 ) ) ⋅ N ( x ; μ 1 ν 2 + μ 2 ν 1 ν 1 + ν 2 , ν 1 ν 2 ν 1 + ν 2 ) \begin{aligned} & \ \ \ \ \mathcal N(x; \mu_1, \nu_1) \mathcal N(x; \mu_2, \nu_2) \\ &= \frac{1}{\sqrt{2 \pi \nu_1}} \exp \left ( - \frac{ (x - \mu_1)^2 }{2 \nu_1} \right ) \cdot \frac{1}{\sqrt{2 \pi \nu_2}} \exp \left ( - \frac{ (x - \mu_2)^2 }{2 \nu_2} \right ) \\ & = \frac{1}{2 \pi \sqrt{ \nu_1 \nu_2}} \exp \left \{ - \frac{ \nu_2 ( x^2 - 2 \mu_1 x + \mu_1^2 ) }{2 \nu_1 \nu_2} - \frac { \nu_1 ( x^2 - 2 \mu_2 x + \mu_2^2 ) }{2 \nu_1 \nu_2} \right \} \\ & = \frac{1}{2 \pi \sqrt{ \nu_1 \nu_2}} \exp \left \{ - \frac{ (\nu_1 + \nu_2) x^2 - 2 (\mu_1 \nu_2 + \mu_2 \nu_1) x + \mu^2_1 \nu_2 + \mu^2_2 \nu_1 }{2 \nu_1 \nu_2} \right \} \\ & = \frac{1}{2 \pi \sqrt{ \nu_1 \nu_2}} \exp \left \{ - \frac{ x^2 - 2 \frac{(\mu_1 \nu_2 + \mu_2 \nu_1)}{\nu_1 + \nu_2} x + \left ( \frac{(\mu_1 \nu_2 + \mu_2 \nu_1)}{\nu_1 + \nu_2} \right)^2 + \frac{\mu^2_1 \nu_2 + \mu^2_2 \nu_1}{\nu_1 + \nu_2} - \left ( \frac{(\mu_1 \nu_2 + \mu_2 \nu_1)}{\nu_1 + \nu_2} \right)^2 }{2 \frac{\nu_1 \nu_2} { \nu_1 + \nu_2 }} \right \} \\ &= \frac{1}{ \sqrt{2 \pi (\nu_1 + \nu_2)} } \exp \left \{ - \frac{\mu^2_1 \nu_2 + \mu^2_2 \nu_1 - \frac {(\mu_1 \nu_2 + \mu_2 \nu_1)^2}{ \nu_1 + \nu_2} } { 2 \nu_1 \nu_2 } \right \} \cdot \frac{1}{\sqrt{2 \pi \frac{\nu_1 \nu_2} { \nu_1 + \nu_2 } }} \exp \left \{ - \frac{ \left( x- \frac{\mu_1 \nu_2 + \mu_2 \nu_1}{\nu_1 + \nu_2} \right)^2 }{ 2 \frac{\nu_1 \nu_2} { \nu_1 + \nu_2 } } \right \} \\ &= \frac{1}{ \sqrt{2 \pi (\nu_1 + \nu_2)} } \exp \left \{ - \frac{ (\mu_1 - \mu_2)^2 }{2 (\nu_1 + \nu_2)} \right \} \cdot \frac{1}{\sqrt{2 \pi \frac{\nu_1 \nu_2} { \nu_1 + \nu_2 } }} \exp \left \{ - \frac{ \left( x- \frac{\mu_1 \nu_2 + \mu_2 \nu_1}{\nu_1 + \nu_2} \right)^2 }{ 2 \frac{\nu_1 \nu_2} { \nu_1 + \nu_2 } } \right \} \\ & = \mathcal N(\mu_1; \mu_2, (\nu_1 + \nu_2)) \cdot \mathcal N(x; \frac{\mu_1 \nu_2 + \mu_2 \nu_1}{\nu_1 + \nu_2}, \frac{\nu_1 \nu_2} { \nu_1 + \nu_2 }) \end{aligned} N(x;μ1,ν1)N(x;μ2,ν2)=2πν11exp(−2ν1(x−μ1)2)⋅2πν21exp(−2ν2(x−μ2)2)=2πν1ν21exp{−2ν1ν2ν2(x2−2μ1x+μ12)−2ν1ν2ν1(x2−2μ2x+μ22)}=2πν1ν21exp{−2ν1ν2(ν1+ν2)x2−2(μ1ν2+μ2ν1)x+μ12ν2+μ22ν1}=2πν1ν21exp⎩ ⎨ ⎧−2ν1+ν2ν1ν2x2−2ν1+ν2(μ1ν2+μ2ν1)x+(ν1+ν2(μ1ν2+μ2ν1))2+ν1+ν2μ12ν2+μ22ν1−(ν1+ν2(μ1ν2+μ2ν1))2⎭ ⎬ ⎫=2π(ν1+ν2)1exp⎩ ⎨ ⎧−2ν1ν2μ12ν2+μ22ν1−ν1+ν2(μ1ν2+μ2ν1)2⎭ ⎬ ⎫⋅2πν1+ν2ν1ν21exp⎩ ⎨ ⎧−2ν1+ν2ν1ν2(x−ν1+ν2μ1ν2+μ2ν1)2⎭ ⎬ ⎫=2π(ν1+ν2)1exp{−2(ν1+ν2)(μ1−μ2)2}⋅2πν1+ν2ν1ν21exp⎩ ⎨ ⎧−2ν1+ν2ν1ν2(x−ν1+ν2μ1ν2+μ2ν1)2⎭ ⎬ ⎫=N(μ1;μ2,(ν1+ν2))⋅N(x;ν1+ν2μ1ν2+μ2ν1,ν1+ν2ν1ν2)
即
N ( x ; μ 1 , ν 1 ) N ( x ; μ 2 , ν 2 ) = N ( μ 1 ; μ 2 , ( ν 1 + ν 2 ) ) ⋅ N ( x ; μ 1 ν 2 + μ 2 ν 1 ν 1 + ν 2 , ν 1 ν 2 ν 1 + ν 2 ) \mathcal N(x; \mu_1, \nu_1) \mathcal N(x; \mu_2, \nu_2) = \mathcal N(\mu_1; \mu_2, (\nu_1 + \nu_2)) \cdot \mathcal N(x; \frac{\mu_1 \nu_2 + \mu_2 \nu_1}{\nu_1 + \nu_2}, \frac{\nu_1 \nu_2} { \nu_1 + \nu_2 }) N(x;μ1,ν1)N(x;μ2,ν2)=N(μ1;μ2,(ν1+ν2))⋅N(x;ν1+ν2μ1ν2+μ2ν1,ν1+ν2ν1ν2)
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