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发帖数: 59 | 1 Let Z1 be N(m1,V1) and let Z2 be N(m2,V2)
Pr(Z1=x|Z=y) \propto Pr(Z1=x,Z=y)
\propto Pr(Z=y|Z1=x)Pr(Z1=x)
\propto 1/((2\pi)^(N/2) |V2|^{1/2}) exp(-(y-x-m2)^T V2^{-1} (y-x-m2))
*1/((2\pi)^(N/2) |V1|^{1/2}) exp(-(x-m1)^T V1^{-1} (x-m1))
Note that the index term in exp function is a quadratic function, so Pr(Z1=x
|Z=y) is proportional to a Normal function. By completing the squares, we
can obtain:
Pr(Z1=x|Z=y) \propto N(m,V),
where V=(V1^{-1}+V2^{-1})^{-1} and m=V(V2^{-1}(y-m2)+V1^{-1}m1) |
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