# Is the bias also a "weight" in a neural network?

I'm learning about how neural networks are trained. I understand how a neuron works, backpropagation, and all that. In neurons, there is a clear distinction between a "weight" and a "bias".

$$Y= \sigma(\text{weight} * \text{input})+ \text{bias}$$

However, all the sources I've found when you train the network you just adjust the weights. Not the bias. However, they never mention what the bias should do, which leads me to think that you just merge all weights and biases in a $$W$$ vector and call it weights, even though there are also biases. Is that correctly understood?

• bias is just a simple kind of weight to shift the separation line, it can be imagined as a weight which is connected to input (original or previous layer) of value 1 always Apr 14 at 8:39

To understand why this works, suppose the outputs of the previous layer are $$u_1, \dots, u_n, u_{n + 1}$$, where $$u_{n + 1}$$ is always $$1$$. Then a neuron in the next layer (without a bias) computes
$$\sigma\left(\sum_{i = 1}^{n + 1} w_i u_i \right) = \sigma\left(\sum_{i = 1}^n w_i u_i + w_{n + 1}\right),$$ where $$\sigma$$ is the activation function. So, the weight $$w_{n + 1}$$ just serves as the bias because it is multiplied by $$u_{n + 1} = 1$$.