Logistic Regression
[Sigmoid Function](https://texonom.com/sigmoid-function-417e17b995cf40c996cec78f699ea59b) [Logit Function](https://texonom.com/logit-function-104e55a12b234813a020cb09ec51c754) [Gaussian Discriminant Analysis](https://texonom.com/gaussian-discriminant-analysis-67ec78f9fd6944e68671c610aeadb079) [Data Classification](https://texonom.com/data-classification-0ed6ffa3555b4879be09f0501f0b580c) 


Sigmoid 함수를 사용하여 0과 1 사이의 값을 출력, log-likelihood를 최대화
![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ff587cf29-0b05-4917-a3f4-6d1add80bbb5%2FUntitled.png?table=block&id=11da7a73-d114-412e-80a4-98503f07fc58&cache=v2)
to make convex cost function
![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F3cd79553-1597-4710-8b7a-e4cf3761cdb6%2FUntitled.png?table=block&id=403a0496-8b25-4ad8-9e9f-40e88eedabea&cache=v2)
### Logistic Regression Notion
|Title|
|:-:|
|[Multi-Class Logistic Regression](https://texonom.com/multi-class-logistic-regression-545d3a376a4d49dd8845111405909327)|
$$h_\theta(x) = g(\theta^Tx) = \frac{1}{1 + e^{-\theta^Tx}}$$
This leads to $p(u|x;\theta)$ described by [Bernoulli Distribution](https://texonom.com/bernoulli-distribution-bddaf7e81c9a4df8ab389521fc56e676) 
$$p(y|x;\theta) = (h_\theta(x))^y(1 - h_\theta(x))^{1 - y}$$
the [Likelihood function](https://texonom.com/likelihood-function-0750726af35a45bb86036dbc7507f725) is
$$l(\theta) = logL(\theta) = \Sigma_{i=1}^ny^{(i)}logh(x^{(i)}) + (1 - y^{(i)})log(1 - h(x^{(i)}))$$
But 로지스틱 회귀의 y값은
you can use [Newton–Raphson method](https://texonom.com/newtonraphson-method-0bc249bfb0fe4e498829110bf785eb6c) or [Stochastic Gradient Descent](https://texonom.com/stochastic-gradient-descent-d8b8d008e0a34f4bb55175ffba21db44) and prior one achieves **faster convergence** or [IRLS](https://texonom.com/irls-ca318bd9d30e4054b87415503e34c0f3) is much faster
difference between [Linear Regression](https://texonom.com/linear-regression-0137f4fba64746f6bc28e059d0aa6dd1) is that usage of sigmoid function or logistic function








