{\mbox{\boldmath $w$}}_{j_H} > 0 \right)\end{aligned}$$ where ${\mbox{\boldmath $X$}}_i \in {\mbox{\boldmath $I$}}^{d_x}$ is the $i$-th input training pattern, ${\mbox{\boldmath $w$}}_{j_k} \in {\mbox{\boldmath $J$}}_k^{d_{k-1}}$ is the weight vector of the hidden neuron $j_k$ in the hidden layer $k$, and $U_l^{(k)}$ is the net input to the hidden neuron $l$ in the hidden layer $k$ (Fig. \[fig5\]).

We prove by mathematical induction that Eq. (\[eqn4-1\]) holds true.

\[For $H=1$\] This case corresponds to a three-layered neural network. It follows that $$\begin{aligned}
\label{eqn4-3}
P\Big( [Z_i]_{(j,p)} = 1 \Big) &=& P ( {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_{j_1} > 0 ) \ \ \ 
(\mbox{from Eq.} (\ref{eqn4-2})) \nonumber\\
&=& \frac{1}{2} \end{aligned}$$ where $j_1$ is a hidden neuron on path $p$. We can see below that the last equality of Eq. (\[eqn4-3\]) holds true. For a given input training pattern ${\mbox{\boldmath $X$}}_i \in {\mbox{\boldmath $I$}}^{d_x}$, $\{ {\mbox{\boldmath $w$}}_{j_1} \in {\mbox{\boldmath $J$}}_1^{d_x} \vert \ {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_{j_1} > 0 \}$ is a half-open hyperspace with a normal vector ${\mbox{\boldmath $X$}}_i$ through the origin in a $d_x$-dimensional Euclidean space (Fig. \[fig6\]). According to the even initialization scheme, the random variables $w_{j_11}, \cdots, w_{j_1d_x}$ which are the components of the weight vector ${\mbox{\boldmath $w$}}_{j1} = (w_{j_11} \cdots w_{j_1d_x})^T$ of the 