$$ 0 \leq h_\theta \leq 1 \ \dots \ g(\theta^Tx) \text{ inside of this is actualy our z which we can plug in} \ g(z) = \frac{1}{1+e^{-z}} \ h_\theta(x) = \frac{1}{1+e^{-\theta^TX}} \text{ output will be probability of y being 1 given x with param theta} $$

$$ \text {probability of a zero is equal to full probability minus probability of one} \\ P(y=0|x;\theta) = 1 - P(y=1|x;\theta) \\ $$

$$ y = 1 \text{ if } h_\theta(x)\gt0.5 \text{ holds true if } \theta^Tx \gt 0 \ \dots \ y = 0 \text{ if } h_\theta(x)\lt0.5 \text{ holds true if } \theta^Tx \lt 0 $$

$$ h_\theta(x) = \frac{1}{1+e^{-\theta^T*X}} \\ $$

$$ \dots \\ \text{think about previous made cost function for logistic regression}\\ \text{Cost}(h_\theta(x), y) \dots \\ \text{however in logistic regression this is non convex (has local minima) because our } h_\theta \text{ is non lineair } $$

$$ \text{cost}(h_\theta(x), y) = \begin{cases} -log(h_\theta(X)) & \text{if } \text{y = 1}\\ -log(1-h_\theta(X)) & \text{if } \text{y = 0} \end{cases} \\ ----\\ \text {now this would be handier to implement if we had a single equation} \\ ----\\ \text{cost}(h_\theta(x), y) = [-y*log(h_\theta(X))] - [(1-y) * log(1-h_\theta(X))] \\ ----\\ --OR--\\ J(\theta) = -\frac{1}{m}[\sum(y^i)logh_\theta(x^{(i)})+(1-y^{(i)})log(1-h_\theta(x^{(i)}))] $$

$$ \text{while } J(\theta) \text{ not minimum } \\ \text{do } \theta_j := \theta_j - \alpha\frac{\delta}{\delta\theta_j}J(\theta) $$

vectorized implementation: $$ x =\begin{bmatrix}x_0 \x_1 \x_2 \x_3 \\end{bmatrix}z^2 =\begin{bmatrix} z_1^{(2)} \z_2^{(2)} \z_3^{(2)} \\end{bmatrix} \z^{2} = \theta^{(1)}*x \text{ is equal to } z^{2} = \theta^{(1)}*a^{(1)} \a^{(2)} = g(z^{2}) \text{ add } a^{(2)}0 = 1 \z^{3} = \theta^{(2)}*a^{(2)} \ah\theta = a^{3} = g(z^{(3)}) $$