N(x|μ1,σ1^2)N(x|μ1,σ2^2) ∝ N(x|μ1+μ2,σ1^2+σ2^2)
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N is d-th dimensional Gaussian distribution. In case of d=1
$ N(x|mu,sigma^2)=frac{1}{sqrt{2pisigma^2}}e^{-frac{(x-mu)^2}{2sigma^2}} $
in case of $d neq 1$,
$N({bf x}|{boldsymbol mu},{bf Sigma})=frac{1}{sqrt{(2pi)^d|{bf Sigma}|}}{rm exp}{-frac{1}{2}(bf{x}-boldsymbol mu)^TSigma^{-1}(x-boldsymbol mu)}$
Here,$bf{x}$ and $boldsymbol mu$ are d dimensional vertical vectors and $bf Sigma$ is d dimensional square matrix.
We can proof $N(x|mu_1,sigma_1)N(x|mu,sigma_2)propto N(x|mu_1+mu_2,sigma_1^2+sigma_2^2)$ easily.
How about in case of production of multi dimensional and single dimensional Gaussian distribution??
I mean, what is $N(x|mu,sigma^2)N({bf W}|{bf m},bfSigma^2) $ proportional to? I want to know what kind of probability distribution will be appeared.
probability statistics
asked Nov 15 at 4:54
Sakurai.JJ
687
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up vote
0
down vote
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N is d-th dimensional Gaussian distribution. In case of d=1
$ N(x|mu,sigma^2)=frac{1}{sqrt{2pisigma^2}}e^{-frac{(x-mu)^2}{2sigma^2}} $
in case of $d neq 1$,
$N({bf x}|{boldsymbol mu},{bf Sigma})=frac{1}{sqrt{(2pi)^d|{bf Sigma}|}}{rm exp}{-frac{1}{2}(bf{x}-boldsymbol mu)^TSigma^{-1}(x-boldsymbol mu)}$
Here,$bf{x}$ and $boldsymbol mu$ are d dimensional vertical vectors and $bf Sigma$ is d dimensional square matrix.
We can proof $N(x|mu_1,sigma_1)N(x|mu,sigma_2)propto N(x|mu_1+mu_2,sigma_1^2+sigma_2^2)$ easily.
How about in case of production of multi dimensional and single dimensional Gaussian distribution??
I mean, what is $N(x|mu,sigma^2)N({bf W}|{bf m},bfSigma^2) $ proportional to? I want to know what kind of probability distribution will be appeared.
probability statistics
asked Nov 15 at 4:54
Sakurai.JJ
687
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
N is d-th dimensional Gaussian distribution. In case of d=1
$ N(x|mu,sigma^2)=frac{1}{sqrt{2pisigma^2}}e^{-frac{(x-mu)^2}{2sigma^2}} $
in case of $d neq 1$,
$N({bf x}|{boldsymbol mu},{bf Sigma})=frac{1}{sqrt{(2pi)^d|{bf Sigma}|}}{rm exp}{-frac{1}{2}(bf{x}-boldsymbol mu)^TSigma^{-1}(x-boldsymbol mu)}$
Here,$bf{x}$ and $boldsymbol mu$ are d dimensional vertical vectors and $bf Sigma$ is d dimensional square matrix.
We can proof $N(x|mu_1,sigma_1)N(x|mu,sigma_2)propto N(x|mu_1+mu_2,sigma_1^2+sigma_2^2)$ easily.
How about in case of production of multi dimensional and single dimensional Gaussian distribution??
I mean, what is $N(x|mu,sigma^2)N({bf W}|{bf m},bfSigma^2) $ proportional to? I want to know what kind of probability distribution will be appeared.
probability statistics
asked Nov 15 at 4:54
Sakurai.JJ
687
N is d-th dimensional Gaussian distribution. In case of d=1
$ N(x|mu,sigma^2)=frac{1}{sqrt{2pisigma^2}}e^{-frac{(x-mu)^2}{2sigma^2}} $
in case of $d neq 1$,
$N({bf x}|{boldsymbol mu},{bf Sigma})=frac{1}{sqrt{(2pi)^d|{bf Sigma}|}}{rm exp}{-frac{1}{2}(bf{x}-boldsymbol mu)^TSigma^{-1}(x-boldsymbol mu)}$
Here,$bf{x}$ and $boldsymbol mu$ are d dimensional vertical vectors and $bf Sigma$ is d dimensional square matrix.
We can proof $N(x|mu_1,sigma_1)N(x|mu,sigma_2)propto N(x|mu_1+mu_2,sigma_1^2+sigma_2^2)$ easily.
How about in case of production of multi dimensional and single dimensional Gaussian distribution??
I mean, what is $N(x|mu,sigma^2)N({bf W}|{bf m},bfSigma^2) $ proportional to? I want to know what kind of probability distribution will be appeared.
probability statistics
probability statistics
asked Nov 15 at 4:54
Sakurai.JJ
687
asked Nov 15 at 4:54
Sakurai.JJ
687
asked Nov 15 at 4:54
Sakurai.JJ
687
asked Nov 15 at 4:54
Sakurai.JJ
687
asked Nov 15 at 4:54
Sakurai.JJ
687
687
add a comment |
add a comment |
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