Question 17: Consider a single-layer neural network with two neurons …

Consider a single-layer neural network with two neurons in the hidden layer. The weight parameters of the network are given as follows:

$w_1^{(1)} = \frac{1}{2},\ w_2^{(1)} = \frac{1}{2}$

$w_1^{(2)} = \frac{1}{2},\ w_2^{(2)} = -\frac{1}{2}$

$w^{out} = \begin{bmatrix} 1 & -1 \end{bmatrix}^T,$

where $w_i^{(j)}$ represents the weight associated with the $j$-th neuron for the $i$-th input feature. Assume that we are solving a binary classification problem.

Based on the above data, answer the given subquestions.

The output layer of the neural network will return a probability $p$ for the input $x_{test} = \begin{bmatrix} 2 & 4 \end{bmatrix}^T$. The sigmoid function is used as the activation function in both the hidden and the output layer of the neural network. Find the value of $p$. Enter the answer correct to two decimal places.