= Zero-inflated Poisson regression
{title2=$\log\mu=x^T\beta,\quad\operatorname{logit}\pi=z^T\gamma$}
A zero-inflated Poisson regression specifies a structural-zero probability $\pi$ and a susceptible-component <Poisson distribution> mean $\mu$. In this parametrization,
$$
P(Y=0)=\pi+(1-\pi)e^{-\mu},\qquad P(Y=y)=(1-\pi)e^{-\mu}\mu^y/y!\quad(y>0).
$$
The mean is $(1-\pi)\mu$. Logistic predictors for $\pi$ and logarithmic predictors for $\mu$ allow covariates to affect component membership and count intensity differently. A susceptible individual can still generate a zero count; susceptibility is not equivalent to an observed positive response.
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