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The logistic regression model,

\begin{equation} \operatorname{p}(X) = \frac{\operatorname{e}^{\beta_0 + \beta_1 X}}{1 + \operatorname{e}^{\beta_0 + \beta_1 X}} \end{equation}

is said to create a decision boundary linear in $X$.

As far as I understand, only the logit is linear in $X$. Is this the reason the decision boundary is linear in X? If so, why? And if this is not the case, what is the reason for this phenomenon?

I am confused about this because the decision boundary can be expressed as: \begin{equation} \operatorname{p}(X) = a, a \in [0,\, 1] \end{equation} And $\operatorname{p}(X)$ is not linear in $X$.

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The decision boundary $p(X)=a, a\in(0,1)$ is

$$\beta_0+\beta_1X=\log\frac{a}{1-a}$$

which is linear in $X$.

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    $\begingroup$ Ah, thanks for showing me the relation of the decision boundary to the logit. The decision boundary is thus a hyperplane, and thus linear in $X$. $\endgroup$ – Nurmister Sep 15 '18 at 16:43

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