How to derive the optimal bayesian solution to a model of two normal distributed populations
In the "Introduction" section of the paper Support-Vector Networks, it mentioned Fisher's solution to a model of two normal distributed populations:
My questions are:
How to derive equation (1)? I even doubt that it should be:
$F_{sq} (x) = sign left [ left ( x - m_1 right ) ^T Sigma_1 ^{-1} left ( x - m_1 right ) - left ( x - m_2 right ) ^T Sigma_2 ^{-1} left ( x - m_2 right ) + ln { dfrac { left | Sigma_1 right |} { left | Sigma_2 right | } } right ]$
because according to Linear discriminant analysis, the solution is:
Why the number of free parameters in equation (1) is $dfrac {n (n + 3)} {2}$?
In my opinion, $m_1, m_2, Sigma_1, Sigma_2$ are all free parameters, because
$Sigma_1, Sigma_2$ are symmetric matrices, so the number should be $n + n + dfrac {n left (n + 1 right )}{2} + dfrac {n left (n + 1 right )}{2} = n left (n + 3 right )$Why the number of free parameters in equation (2) is $n$?
We can rewrite Equation (2) as $F_{sq} left ( X right ) = WX + b$, so $W$ and $b$ are both free parameters, then the number should be $n + 1$.
optimization bayesian fisher-information
asked Nov 26 at 15:54
Jiongjiong Li
225
add a comment |
In the "Introduction" section of the paper Support-Vector Networks, it mentioned Fisher's solution to a model of two normal distributed populations:
My questions are:
How to derive equation (1)? I even doubt that it should be:
$F_{sq} (x) = sign left [ left ( x - m_1 right ) ^T Sigma_1 ^{-1} left ( x - m_1 right ) - left ( x - m_2 right ) ^T Sigma_2 ^{-1} left ( x - m_2 right ) + ln { dfrac { left | Sigma_1 right |} { left | Sigma_2 right | } } right ]$
because according to Linear discriminant analysis, the solution is:
Why the number of free parameters in equation (1) is $dfrac {n (n + 3)} {2}$?
In my opinion, $m_1, m_2, Sigma_1, Sigma_2$ are all free parameters, because
$Sigma_1, Sigma_2$ are symmetric matrices, so the number should be $n + n + dfrac {n left (n + 1 right )}{2} + dfrac {n left (n + 1 right )}{2} = n left (n + 3 right )$Why the number of free parameters in equation (2) is $n$?
We can rewrite Equation (2) as $F_{sq} left ( X right ) = WX + b$, so $W$ and $b$ are both free parameters, then the number should be $n + 1$.
optimization bayesian fisher-information
asked Nov 26 at 15:54
Jiongjiong Li
225
add a comment |
In the "Introduction" section of the paper Support-Vector Networks, it mentioned Fisher's solution to a model of two normal distributed populations:
My questions are:
How to derive equation (1)? I even doubt that it should be:
$F_{sq} (x) = sign left [ left ( x - m_1 right ) ^T Sigma_1 ^{-1} left ( x - m_1 right ) - left ( x - m_2 right ) ^T Sigma_2 ^{-1} left ( x - m_2 right ) + ln { dfrac { left | Sigma_1 right |} { left | Sigma_2 right | } } right ]$
because according to Linear discriminant analysis, the solution is:
Why the number of free parameters in equation (1) is $dfrac {n (n + 3)} {2}$?
In my opinion, $m_1, m_2, Sigma_1, Sigma_2$ are all free parameters, because
$Sigma_1, Sigma_2$ are symmetric matrices, so the number should be $n + n + dfrac {n left (n + 1 right )}{2} + dfrac {n left (n + 1 right )}{2} = n left (n + 3 right )$Why the number of free parameters in equation (2) is $n$?
We can rewrite Equation (2) as $F_{sq} left ( X right ) = WX + b$, so $W$ and $b$ are both free parameters, then the number should be $n + 1$.
optimization bayesian fisher-information
asked Nov 26 at 15:54
Jiongjiong Li
225
In the "Introduction" section of the paper Support-Vector Networks, it mentioned Fisher's solution to a model of two normal distributed populations:
My questions are:
How to derive equation (1)? I even doubt that it should be:
$F_{sq} (x) = sign left [ left ( x - m_1 right ) ^T Sigma_1 ^{-1} left ( x - m_1 right ) - left ( x - m_2 right ) ^T Sigma_2 ^{-1} left ( x - m_2 right ) + ln { dfrac { left | Sigma_1 right |} { left | Sigma_2 right | } } right ]$
because according to Linear discriminant analysis, the solution is:
Why the number of free parameters in equation (1) is $dfrac {n (n + 3)} {2}$?
In my opinion, $m_1, m_2, Sigma_1, Sigma_2$ are all free parameters, because
$Sigma_1, Sigma_2$ are symmetric matrices, so the number should be $n + n + dfrac {n left (n + 1 right )}{2} + dfrac {n left (n + 1 right )}{2} = n left (n + 3 right )$Why the number of free parameters in equation (2) is $n$?
We can rewrite Equation (2) as $F_{sq} left ( X right ) = WX + b$, so $W$ and $b$ are both free parameters, then the number should be $n + 1$.
optimization bayesian fisher-information
optimization bayesian fisher-information
asked Nov 26 at 15:54
Jiongjiong Li
225
asked Nov 26 at 15:54
Jiongjiong Li
225
asked Nov 26 at 15:54
Jiongjiong Li
225
asked Nov 26 at 15:54
Jiongjiong Li
225
asked Nov 26 at 15:54
Jiongjiong Li
225
225
add a comment |
add a comment |
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