# 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:

1. 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:

2. 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 $$n + n + \dfrac {n \left (n + 1 \right )}{2} + \dfrac {n \left (n + 1 \right )}{2} = n \left (n + 3 \right )$$

3. 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 $$n + 1$$.

• I think that you should split your post into different questions, so it is easier to help you. – nicoguaro Nov 26 '18 at 16:05
• Will better answers be given on the stats stackexchange? – user14717 Dec 25 '18 at 18:02

2. To get a degenerate case $$w.x+b$$ from the difference of the two quadratics, we must have $$\Sigma_1=\Sigma_2=\Sigma$$, hence there are $$m_1, m_2, \Sigma$$ free parameters $$n+n+n(n-1)/2=n(n+3)/2$$
• Thanks a lot, but I'm still confused: 1. The paper " Why the logistic function?" derived Equation (2) for the condition $\Sigma_1 = \Sigma_2$ , rather than quadratic equation (1). 2. paper "Support-Vector Networks" actually mentioned that the number of free parameters for Equation (2) is $n$, as is shown in the green blocks in my post. – Jiongjiong Li Nov 25 '18 at 13:42