This paper provides a practical simulation-based Bayesian analysis of parameter-driven models for time series Poisson data with the AR(1) latent process. The posterior distribution is simulated by a Gibbs sampling algorithm. Full conditional posterior distributions of unknown variables in the model are given in convenient forms for the Gibbs sampling algorithm. The case with missing observations is also discussed. The methods are applied to real polio data from 1970 to 1983.