Why is the formula for the logistic regression not shifted in the x-axis by 0.5 as follows?

f(x)=1/(1+e^-(wx+b-0.5)) ?

The reason for me thinking that we should add that “-0.5” is so that the value of the the sigmoid function and the value of linear function wx+b are equal at y = 0.5. (since 0.5 is half way between the two possible target values 0 and 1 in the data set)

So when the linear function predicts a value less than 0.5 then the sigmoid function predicts a value less than 0.5 and vice versa.

The problem I’m seeing with the default sigmoid function is that when the linear function is predicting a value between 0 and 0.5 (hence “False” output), the sigmoid function will predict a value greater than 0.5 (hence “True” output).

What am I missing?

The decision boundary is at f(x) = 0.5, because that’s the midpoint of the sigmoid function (which ranges from 0 to 1).

It is not the value of the linear function that is directly used to make the decision: it is the output of sigmoid. So the linear function output being between 0 and 0.5 does not predict “False”. The point is that `sigmoid(0) = 0.5`

and sigmoid is monotonic. So if the linear value is \leq 0, then sigmoid will be \leq 0.5 and we interpret that as predicting “False”. If the linear output is > 0 then sigmoid > 0.5 will predict “True”. We are interpreting the sigmoid output as being the probability of “True”.

The thing some people have suggested is “why don’t we use `tanh`

instead of `sigmoid`

and then use` tanh(z) > 0`

as `True`

”. That is discussed on this thread.