Can a neuron have both a bias and a threshold?

Question

I have not seen a neuron that uses both a bias and a threshold. Why is this?

Answer 1

I assume you're talking about a perceptron threshold function. One definition of it with an explicit threshold is $$f(\textbf{x})= \begin{cases} 1& \text{if } \textbf{w}\cdot\textbf{x} > t\\ 0& \text{otherwise} \end{cases}.$$

Another form with a bias is $$f(\textbf{x})= \begin{cases} 1& \text{if } \textbf{w}\cdot\textbf{x} + b > 0\\ 0& \text{otherwise} \end{cases}.$$

But these forms are of course equivalent if you set $b=-t$.

There's nothing stopping you from using a perceptron definition with both a bias and a threshold: $$f(\textbf{x})= \begin{cases} 1& \text{if } \textbf{w}\cdot\textbf{x} +b > t\\ 0& \text{otherwise} \end{cases}.$$

But this is also equivalent to the other two forms. We can rewrite this as $$f(\textbf{x})= \begin{cases} 1& \text{if } \textbf{w}\cdot\textbf{x} > t'\\ 0& \text{otherwise} \end{cases}$$ where $t'=t-b$ is the new threshold. Or, we could rewrite it as $$f(\textbf{x})= \begin{cases} 1& \text{if } \textbf{w}\cdot\textbf{x} + a> 0\\ 0& \text{otherwise} \end{cases}$$ where $a=b-t$ is the new bias.

You never see definitions of this function with both a threshold and a bias because it has simpler forms.