### Summary

The logit function is fundamental to many machine learning algorithms, primarily used in logistic regression. It’s a mathematical method that translates a variable into a range between 0 and 1, hence suitable for binary classification problems.

### ELI5

Imagine you and your friend are playing a guessing game. You need to guess a number your friend is thinking of, but the only hint you’re given is whether you’re too low or too high. The logit function is like a rule in the game that does that for numbers, but instead of guessing, it can squish any number into a box from 0 to 1.

### In-depth explanation

The logit function, log-odds, or the logit model, is a function that maps probability ranges from (0, 1) to (-∞, ∞). It’s typically used in binary logistic regression models, where the goal is to find the probability of a certain class or event to occur.

`logit(p) = ln(p/(1-p))`

Here, `p` is the probability of positive events. The logit function is the logarithm of odds of `p`. If `p` denotes probability, then `p/(1-p)` is the odds, and `ln(p/(1-p))` is the logarithm of odds—that is, the logit. The logit function is the inverse of the sigmoid function used in logistic regression.

In a logistic regression, you are essentially trying to find the ’line of best fit’ through the binary datapoints, but since ’line’ doesn’t make sense in a binary world, you need a function which indicates probability. Enter the sigmoid function—it converts the line to a curve ranging from 0 to 1, and is excellent for indicating probabilities.

The output of the sigmoid function is then fed back into the inverse function, i.e., the logit function, during the learning process (backpropagation), to update the weights for the next iterations.

Python code for logit function would look something like this:

import numpy as np

def logit(p): return np.log(p / (1 - p))

But in ML, you’ll often use a library that has these functions embedded—like scikit-learn’s LogisticRegression or TensorFlow/Keras for neural networks.

Sigmoid Function, Binary Logistic Regression, Probabilities, Odds, Binary Classification, Likelihood Functions, Backpropagation, Neural Networks.