Posterior structural-zero probability ($P(Z=1\mid Y=0)=\frac{\pi }{\pi +(1-\pi )e^{-\mu }}$)

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.