So correct me if i'm wrong.
The usual link function for poisson regression is log, so that you're performing regression on log(y)~x1+x2+x3+x4+…
The variable y is typically a count, meaning it is restricted to integers from 0 to positive infinity. The input variables x1…xn are not restricted to the positive integers.
So how does the regression proceed when y = 0?
Is log(0) merely ignored?
Also, to be clear, this question is not about zero-inflated poisson regression (which distinguishes between different kinds of zeros).
Best Answer
The Poisson model is $$y = \exp \left(\alpha + \beta \cdot x + \varepsilon \right).$$
The way you get an outcome of zero is when the index $\alpha + \beta \cdot x + \varepsilon$ is large and negative. The coefficients do not come from a regression of logged outcome on the covariates, but from maximization of the log likelihood. You can also use this model on non-integer outcomes, though that is more controversial.
You can learn more about this model from this blog post, including the zeros issue and a comparison to logged outcome regression.