# Logistic Regression
It is a **classification** algorithm used to estimate discrete values ( binary values like 0/1, yes/no, true/false ) based on given set of independent variable(s). In simple words, it predicts the probability of occurrence of an event by fitting data to a logit function.
Hence, **logistic regression** is also known as **logit regression**. Since, it predicts the probability, its output values lies between 0 and 1 (as expected).
Let’s say your friend gives you a puzzle to solve. There are only 2 outcome scenarios – either you solve it or you don’t.Coming to the math, the log odds of the outcome is modeled as a linear combination of the predictor variables.
```
odds= p/ (1-p) = probability of event occurrence / probability of not
\event occurrence
ln(odds) = ln(p/(1-p))
logit(p) = ln(p/(1-p)) = b0+b1X1+b2X2+b3X3....+bkXk
```
Above, p is the probability of presence of the characteristic of interest. *It chooses parameters that maximize the likelihood of observing the sample values rather than that minimize the sum of squared errors (like in ordinary regression).*
Now, you may ask, why take a log? For the sake of simplicity, let’s just say that this is one of the best mathematical way to replicate a step function.
### What if we just use linear regression on discrete classes?
**Pink**: a workable case
**Blue**: a problematic case
### **Linear Regression vs Logistic Regression output value range:**
### Logistic Function
A **logistic function** or **logistic curve** is a common "S" shape ([sigmoid curve](https://en.wikipedia.org/wiki/Sigmoid_function)), with equation:
where
* e = the [natural logarithm](https://en.wikipedia.org/wiki/Natural_logarithm) base (also known as [Euler's number](https://en.wikipedia.org/wiki/E_\(mathematical_constant\))),
* x0 = the x-value of the sigmoid's midpoint,
* L = the curve's maximum value, and
* k = the steepness of the curve.
### Not use least square cost function since it would be non-convex
### Two class cross entropy loss
h(x) is a binary class logistic function and the output of two classes are already normalized.
### Multi-Class one vs all
### Multinomial Logistic Regression - SoftMax
