EM algorithm for zero-inflated Poisson regression (${\tau }_i=P(Z_i=1\mid Y_i)$)

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.