# Is bias nothing but perceptron threhold value?

I was revisiting neural network basics from this post. The perceptron follows below equation:

\begin{align} y & = 1 & \text{if } \sum_{i=1}^n w_i\times x_i \geq \theta \\ & = 0 & \text{if } \sum_{i=1}^n w_i\times x_i \lt \theta \\ \end{align}

Absorbing threshold value $$\theta$$ to left hand side, we get:

\begin{align} y & = 1 & \text{if } \sum_{i=1}^n w_i\times x_i - \theta \geq 0 \\ & = 0 & \text{if } \sum_{i=1}^n w_i\times x_i - \theta \lt 0 \\ \end{align}

Further including $$\theta$$ in weights and inputs, i.e. $$x_0=1$$ and $$w_0=-\theta$$ and starting sum from $$i=0$$ instead of $$i=1$$, we get

\begin{align} y & = 1 & \text{if } \sum_{\color{red}{i=0}}^n w_i\times x_i \geq 0 \\ & = 0 & \text{if } \sum_{\color{red}{i=0}}^n w_i\times x_i \lt 0 \\ \end{align}

I knew this convention of including bias as a first value in weight vector, but it is first time I am finding "threshold value $$\theta$$" getting moved in as "bias weight $$w_0$$". Is this what always / normally happens? (that is, does all online articles and book mean the same: including threshold as bias. I never knew bias had any connection with threshold value and was always felt that it is independently learnt during training)