> ## Documentation Index
> Fetch the complete documentation index at: https://ai.teamjaaf.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Logistic Regression

> Binary classification using the logistic function.

# 🔐 Logistic Regression

Logistic regression is a linear model for **binary classification**. It estimates the probability that an input $\mathbf{x}$ belongs to the positive class.

## Hypothesis

The model applies the logistic (sigmoid) function to a linear combination of the inputs:

$$
h_{\boldsymbol{\theta}}(\mathbf{x}) = \sigma(\mathbf{x}^T\boldsymbol{\theta}) = \frac{1}{1 + e^{-\mathbf{x}^T\boldsymbol{\theta}}}.
$$

## Cost Function

Parameters are learned by minimizing the **logistic loss**:

$$
J(\boldsymbol{\theta}) = -\frac{1}{m} \sum_{i=1}^m \big[ y^{(i)} \log h_{\boldsymbol{\theta}}(\mathbf{x}^{(i)}) + (1 - y^{(i)}) \log (1 - h_{\boldsymbol{\theta}}(\mathbf{x}^{(i)})) \big].
$$

## Example (scikit-learn)

```python theme={null}
import numpy as np
from sklearn.linear_model import LogisticRegression

X = np.array([[0], [1], [2], [3]])
y = np.array([0, 0, 1, 1])

model = LogisticRegression()
model.fit(X, y)

proba = model.predict_proba([[1.5]])
print(proba)
```

## Interpretation

* Outputs a probability between 0 and 1.
* Decision boundary at $h_{\boldsymbol{\theta}}(\mathbf{x}) = 0.5$.
* For multi-class problems, use **one-vs-rest** or a **softmax regression** extension.