Logistic Regression

Say the data has the following form: $f(x)=g(w_0{x}_0+w_1{x}_1+b)$ So the model is represented as: f_{\mathbf{w},b}(\mathbf{x}^{(i)}) = g(\mathbf{w} \cdot \mathbf{x}^{(i)} + b) \tag{1} and the sigmoid function is: $g(z)=\frac{1}{1+e^{-z}}$ and $\mathbf{w}\cdot \mathbf{x}$ is the vector dot product: $\mathbf{w}\cdot \mathbf{x}=w_0{x}_0+w_1{x}_1$

• We interpret the output of the model ($f_{\mathbf{w},b}(x)$) as the probability that $y=1$ given $\mathbf{x}$ and parameterized by $\mathbf{w}$ and $b$.

• Therefore, to get a final prediction ($y=0$ or $y=1$) from the logistic regression model, we can use the following heuristic -

  if, $f_{\mathbf{w},b}(x)>=0.5$, predict $y=1$ if, $f_{\mathbf{w},b}(x)<0.5$, predict $y=0$

Decision Boundary Logistic Regression Cost Function