(\mbox{from Eq.} (\ref{eqn4-2})) \nonumber\\
& & (\mbox{$w_{j_{H+1} j_H} = \lambda_{H+1}$ is removed because it is independent of 
$[Z_i]_{(j,p)}$}) \nonumber\\
&\fallingdotseq& \hspace*{-0.3cm} \frac{1}{2^H}. 
$$ We prove below by mathematical induction that the approximate equality $(\fallingdotseq)$ in Eq. (\[eqn4-6\]) holds true.

\[For $H=1$\] This case corresponds to a three-layered neural network. For the sake of simplicity, we let ${\mbox{\boldmath $X$}}_i = (x_1 \cdots x_{j_0} \cdots x_{d_0})^T$ and ${\mbox{\boldmath $w$}}_{j_1} = (w_1 \cdots w_{j_0} \cdots w_{d_0})^T$ where $w_{j_0} = w_{j_1 j_0}$. Then, $$\begin{aligned}
\label{eqn4-7}
& & P \left( {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_{j_1} > 0 \ \Big\vert \ w_{j_1 j_0} = \lambda_1\right) \nonumber\\
&=& P \left(x_1 w_1 + \cdots + x_{j_0} w_{j_1 j_0} + \cdots + x_{d_0} w_{d_0} > 0 \ 
\Big \vert \ w_{j_1 j_0} = \lambda_1\right) \nonumber\\
&=& P \left(x_1 w_1 + \cdots + x_{j_0} \lambda_1 + \cdots + x_{d_0} w_{d_0} > 0 \ 
\Big \vert \ w_{j_1 j_0} = \lambda_1\right) \nonumber\\
&=& P \Big(x_1 w_1 + \cdots + x_{j_0} \lambda_1 + \cdots + x_{d_0} w_{d_0} > 0 \ \Big) 
\nonumber\\
& & (\mbox{$w_{j_1 j_0} = \lambda_{1}$ is removed because it is independent of 
the other weights}) \nonumber\\
&=& P \Big(x_1 w_1 + \cdots + x_{j_0 - 1} w_{j_0 -1} + x_{j_0 