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Given an unknown classifier, could someone make assumptions for the linearity of this model from the fact that

$$y=\left\{ \begin{aligned} 1 &\;\mathrm{if}\;& p(y=1\,|\,x) \geq 0.5 \\ -1 &\;\mathrm{otherwise} \end{aligned} \right.$$ ?

Is that the reason we conclude logistic regression is linear?

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No, many nonlinear equations could satisfy $y=\left\{ \begin{aligned} 1 &\;\mathrm{if}\;& p(y=1\,|\,x) \geq 0.5 \\ -1 &\;\mathrm{otherwise} \end{aligned} \right.$

The logistic model can be represented as a regular linear equation except the left-hand side of the equation is transformed through a link function. So for logistic regression, you have $\ln \left( \frac{\hat{p}}{1-\hat{p}} \right) = B_0 + B_1 X$, where $\hat{p}$ is the probability that $Y=1$. You see the coefficients on the right-hand side ($B_0$ and $B_1$) enter the model linearly just like in simple linear regression. The left-hand side of the equation represents the logit function of $\hat{p}$, or the log odds. So you could could intuitively interpret that we are modeling the log odds using simple linear regression. It is only when we back-transform the predictions that we get the nice logistic curve that does not look linear.

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