The Likelihood Function

In our regular likelihood function we would have $f(x_i\mid \theta )$.

With Generalized Linear Models we have $f(x_i\mid {\theta }_i(\beta ))$. This means that each observation has its own parameters (set of parameters).

For example, suppose we have a random variable which follows a Poisson distribution. Denote this random variable as $Y_i$ with parameter ${\lambda }_i$.

Suppose we can approximate the rate for each observation $Y_i$ with some relationship like ${\lambda }_i=exp({\beta }_0+{\beta }_{1}x)$ for some $x_i$ associated with each ${\lambda }_i$.

Then, our likelihood function would be as follows

$$L({\beta }_0,{\beta }_1)=\frac{e^{-({\beta }_0+{\beta }_{1}x)}({\beta }_0+{\beta }_{1}x{)}^{y_i}}{y_i!}$$