Question-19

Consider that you have trained a logistic regression model on a dataset for a binary classification problem. The model predicts the probability of a point being classified as \(1\) based on two features, \(x_1\) and \(x_2\). After training, the model’s decision boundary is defined by the equation \(w_0 + w_1 x_1 + w_2 x_2 = 0\). If the model’s predicted probability for a given point \((x_1, x_2)\) is 0.7, what is the value of \(w_0 + w_1 x_1 + w_2 x_2\) at this point?

• \(\log(0.7)\)

• \(\log(0.7 / 0.3)\)

• \(\log(0.3 / 0.7)\)

• \(-\log(0.7)\)

Answer

• \(\log(0.7)\)

• \(\log(0.7 / 0.3)\)

• \(\log(0.3 / 0.7)\)

• \(-\log(0.7)\)

Solution

We have:

\[ \begin{equation*} \begin{aligned} \frac{1}{1+e^{-\mathbf{w}^{T}\mathbf{x}}} & =p\\\\ \frac{1-p}{p} & =e^{-\mathbf{w}^{T}\mathbf{x}}\\\\ \mathbf{w}^{T}\mathbf{x} & =\ln\left(\frac{p}{1-p}\right) \end{aligned} \end{equation*} \]

For \(\displaystyle p=0.7\), we have:

\[ \begin{equation*} \mathbf{w}^{T} \mathbf{x} = \ln\left(\frac{0.7}{0.3}\right) \end{equation*} \]