Zero-inflated Poisson regression ($\log \mu =x^T\beta ,\mathrm{logit}\pi =z^T\gamma$)

A zero-inflated Poisson regression specifies a structural-zero probability $\pi$ and a susceptible-component Poisson distribution mean $\mu$. In this parametrization,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.

In a zero-inflated Poisson regression, Bayes theorem separates structural zeros from susceptible zero counts. A positive count rules out the structural-zero class; a zero raises its posterior probability according to the displayed formula, but usually does not determine class membership with certainty. This same probability supplies the E-step of the EM algorithm for zero-inflated Poisson regression.

A logarithmic count-component contrast $b$ multiplies the susceptible mean by $e^b$, while a zero-logit contrast $g$ multiplies the structural-zero odds by $e^g$. If the baseline zero predictor is $\eta$, the marginal mean ratio isThus the count ratio alone is not a population-average effect when a covariate changes both components. With interaction terms, the relevant count contrast must first include the interactions for the stated reference group.

The expectation-maximization algorithm introduces $Z_i=1$ for a structural zero. Its E-step gives ${\tau }_i=0$ at positive counts and ${\tau }_i={\pi }_i/[{\pi }_i+(1-{\pi }_i)e^{-{\mu }_i}]$ at zero. The M-step maximizes a logistic log-likelihood with fractional responses ${\tau }_i$ and a Poisson log-likelihood with weights $1-{\tau }_i$. With only a binary predictor, write $n_j$ for group size, $S_j={\sum }_{x_i=j}{\tau }_i$ and $C_j={\sum }_{x_i=j}{Y}_i$; the explicit updates are ${\pi }_j=S_j/n_j$ and ${\mu }_j=C_j/(n_j-S_j)$. Log and logit group contrasts recover the regression coefficients, with boundary values interpreted through limits.