model initialized by the even initialization method, random variable $[Z_i]_{(j,p)}$ is independent of the sequence of weights on path $p$. [$\Box$]{}

[*Proof*]{}.   Take training pattern $i$, output neuron $j$, and path $p$ from an input neuron to output neuron $j$ of the Kawaguchi model arbitrarily and fix them. Denote by $j_0, \cdots, j_H$ the neurons on path $p$ where $j_k$ is the neuron in the $k$-th layer $(k=0, \cdots, H$). Let also $w_{j_1 j_0}, \cdots, w_{j_{H+1} j_H }$ denote $H+1$ weights on path $p$ where $w_{j_{k+1} j_k }$ is the weight between the layer $k$ and the layer $k+1$ on path $p$ ($k=0, \cdots, H$) (Fig. \[fig7\]). Then, for any $\lambda_1, \cdots, \lambda_{H+1} \in {\mbox{\boldmath $R$}}$, $$\begin{aligned}
\label{eqn4-6}
& & \hspace*{-0.3cm} P \left( [Z_i]_{(j,p)} = 1 \ \Big\vert \ w_{j_1 j_0} = \lambda_1, 
\cdots, w_{j_{H+1} j_H } = \lambda_{H+1} \right) \nonumber\\
&=& \hspace*{-0.3cm} P \left( {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_{j_1} > 0, \left[ \varphi( U_1^{(1)} ) \cdots \varphi(U_{d_1}^{(1)} ) \right] {\mbox{\boldmath $w$}}_{j_2} > 0, \cdots, \left[ \varphi( U_1^{(H-1)} ) \cdots \varphi(U_{d_{H-1}}^{(H-1)}) \right] 
{\mbox{\boldmath $w$}}_{j_H} > 0 \right. \nonumber \\
& & \hspace*{5.5cm} \left. \Big\vert \ w_{ j_1 j_0 } = \lambda_1, \cdots, 
w_{ j_{H} j_{H-1} } = \lambda_{H} \right) \ \ 