Logistic Regression With Hypothesis Representation (CS229)

Logistic Regression Model

Goal: $0\le h_{\theta}(x)\le1$

$$h_{\theta}(x)=g(\mathbf{\theta^\top x})$$
where $g(z)$ is a sigmoid function (i.e. logistic function)
$$g(\mathbf{z})=\frac{1}{1+e^{-z}}$$

$$h_{\theta}(x)=p(y=1|_{x;\theta})$$

The equation above means the probability that $y=1$, given $x$, parameterized by $\theta$.

$$p(y=0|_{x;\theta})+p(y=1|_{x;\theta})=1$$
then we obtain the following equation:
$$p(y=0|_{x;\theta})=1-p(y=1|_{x;\theta})$$

The explanation of above equations, for example, the summation chance of malignant tumor $p(y=1|_{x;\theta})$ and the chance of benign(unmalignant) tumor $p(y=0|_{x;\theta})$ is $100\%$ (i.e. $1$).