Poisson Regression - Regression Models

Regression Models

If is a vector of independent variables, then the model takes the form

where and . Sometimes this is written more compactly as

where x is now an (n + 1)-dimensional vector consisting of n independent variables concatenated to some constant, usually 1. Here θ is simply a concatenated to b.

Thus, when given a Poisson regression model θ and an input vector, the predicted mean of the associated Poisson distribution is given by

If Y_i are independent observations with corresponding values x_i of the predictor variable, then θ can be estimated by maximum likelihood. The maximum-likelihood estimates lack a closed-form expression and must be found by numerical methods. The probability surface for maximum-likelihood Poisson regression is always convex, making Newton–Raphson or other gradient-based methods appropriate estimation techniques.