Predicting with a GLM I wanted to check my understanding of predicting with a GLM:
A binomial/logistic regression model predicts the binomial parameter = p = P(success). To convert the probability into classes, we have to add a threshold or cutoff.
The same idea applies for a multinomial logistic regression model.
A poisson regression model predicts the poisson parameter = rate. To convert into counts, I use a threshold again?
A gamma model predicts the scale and rate parameters. I do not need a threshold because the response is continuous.

The commonality is that all these models predict conditional expectations. If your target class is coded 1 and your nontarget class is coded 0, then a predicted probability $\hat{p}$ of a new instance to belong to the target class is just the conditional expectation of the new instance's code. (Thresholding is iffy and loses a lot of information. Only do it if you know what you are doing.)
Your Poisson regression will also predict the conditional expectation. (As long as your prediction is on the response scale, not the linear scale.) You can feed this predicted expectation $\hat{\lambda}$ into a Poisson calculator to get predicted probabilities of each possible count. Note that this procedure is a shortcut that completely disregards the uncertainty in your estimate of $\hat{\lambda}$ - take a look here for a more stringent approach.
Note that there are other models that predict other functionals of the target variable's distribution, e.g., quantile regression, which aims at predicting a certain quantile.
