Count modelling is an increasingly important area of research in applied statistics. Classic count models do not satisfy the demand for problems of dispersion. The rise of Generalized Poisson distribution and Zero Inflated series distributions innovates a proliferation of studies. However, questions have been raised about the intimate connections and relationships among the Generalized Poisson distribution and Zero Inflated series distributions. A considerable amount of literature has been published on the comparison either among the Generalized Poisson distribution, Negative Binomial distribution and Poisson distribution or among Zero Inflated series distributions from a frequentist perspective. In this thesis, we critically evaluate whether the Generalized Poisson model, Zero Inflated Poisson model, Zero Inflated Negative Binomial model are interchangeable or not in exceptional circumstances from a Bayesian viewpoint. The objective is to compare estimated predictions of the mean with three models. To implement our goal, one Metropolis-Hastings algorithm is proposed. We perform a simulation study with data having Generalized Poisson distributions and then apply the Generalized Poisson model, Zero Inflated Poisson model, Zero Inflated Negative Binomial model to the same data set. Posterior distributions of the expectation are obtained based on data set and non-informative priors. Point estimates of the expectation with the Highest Posterior Density interval are calculated and play an important role in model selection together with Cross Validation and posterior predictive checks. Finally, this thesis attempts to demonstrate that the Zero Inflated Poisson model and Zero Inflated Negative Binomial model are superior to the Generalized Poisson model when data with a Generalized Poisson distribution has massive zeros. Forasmuch the dramatic impact of COVID-19, researchers are more concerned about the confirmed cases, but inflated deaths toll also deserves some attention. For this reason, we investigate mortality in two provinces of Canada, Manitoba and Saskatchewan, over the past year. The objective is to provide a well fitting count model for predicting the number of daily deaths. Identical models and approaches in the simulation study are used in the application.