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I have this mass of probability is proportional to $$exp[-\frac{\beta}{2}\parallel x-Hz\parallel^{2}-\frac{\alpha}{2}z^{t}C^{t}Cz]$$ and I have to calculate to its distribution but I don't know how, its distribution is $$\mathcal{N}((\alpha C^{t}C+\beta H^{t}H)^{-1}\beta H^{t} x,(\alpha C^{t}C+\beta H^{t}H)^{-1}.$$ $z$ and $x$ are vectors and $H$ and $C$ matrices, Any idea?

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    $\begingroup$ should the $||x-Hz||$ be $||x-Hz||^2$? And are you familiar with completing the square for vectors? $\endgroup$
    – jld
    Dec 6 '17 at 22:17
  • $\begingroup$ Could you explain your notation a bit more. what is $z$ for instance? $\endgroup$
    – Josh
    Dec 6 '17 at 22:24
  • $\begingroup$ For sure is squared but I don't know how to complete square for vectors. And x and z are a vectors, C and H are matrixs. $\endgroup$ Dec 6 '17 at 22:30
  • $\begingroup$ matrices, not matrixs $\endgroup$
    – Brad S.
    Dec 6 '17 at 23:02
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    $\begingroup$ Do you know matrix algebra? You should expand $\Vert x-Hz \Vert^2$ as$$(x-Hz)^\text{T}(x-Hz)$$ $\endgroup$
    – Xi'an
    Dec 7 '17 at 15:26