Data Science

When we build a model using parameteric regression, we are typically predicting the values of some response variable given some predictor variables
A generalized linear model (or GLM) assumes the conditional distribution of the response variable conditional on predictors $Y|X$ is a member of some family of distribution
Therefore, out model's predictions of $Y|X$ will belong to the same family of distribution
Linear regression assumes the conditional distribution of $Y|X$ is a gaussian distribution
Logistic regression assumes the conditional distribution of $Y|X$ is a bernoulli distribution
Poisson regression assumes the conditional distribution of $Y|X$ is a poisson distribution

A generalized linear model is a type of model where the model's predictions are transformed to some output that is linearly related to the predictors
A generalized linear model is able to achieve this because it assumes the distribution of a given model's predictions (and thus $Y|X$ ) is a member of some family of distribution
The transformation is performed using a link function

As mentioned previously, a link function maps a model's predictions to some transformed output that is linearly related to the predictors
In other words, a link function transforms our model's predictions from a particular distribution to another distribution
A link function achieves this by adjusting the magnitude of the variance of each measurement to be a function of its predicted value
In other words, a link function will map the model's predictions to a scale where the relationship between the transformed predictions and the predictors is linear
This scale is pre-determined according to the the assumed family of distribution for $Y|X$ in the model

The logit function is an example of a link function
If we input a logistic regression model's predictions into the logit function, then the function will return the log-odds
The log-odds are linearly related to the predictors
The logit link function is used to transform probabilities on the probability scale (i.e. $[0,1]$ ) to log-odds on the logit scale
In other words, the logit link function maps probabilities to a scale where the relationship between the transformed probabilities and the predictors is linear

The log function is an example of a link function
If we input a poisson regression model's predictions into the log function, then the function will return some transformed output (related to the incident ratio)
This transformed output is linearly related to the predictors

Linear Regression

Gradient Boosting