Poisson regression is used when it is believed that the target variable is generated from an underlying Poisson distribution, which occurs in cases where the data is measured in counts. The Poisson is a discrete distribution used to model the average number of occurrences of an event over a specified time frame, such as the number of customers that enter a store each hour of the day. As count data cannot be less than zero, GLM uses a log-link function to constrain predictions to be non-negative through the following formulation:
where the expected value of the ith observation is represented by the poisson parameter ƛi
Poisson regression uses maximum likelihood estimation to find the values for the coefficients that maximize the chance of observing the given data if it was generated from a poisson distribution having an underlying parameter ƛ. Inference then proceeds in an analogous way to logistic regression.