Default Priors in a Zero-Inflated Poisson Distribution: Intrinsic Versus Integral Priors

Prior elicitation is an important issue in both subjective and objective Bayesian frameworks, where prior distributions impose certain information on parameters before data are observed. Caution is warranted when utilizing noninformative priors for hypothesis testing or model selection. Since noninformative priors are often improper, the Bayes factor, i.e., the ratio of two marginal distributions, is not properly determined due to unspecified constants contained in the Bayes factor. An adjusted Bayes factor using a data-splitting idea, which is called the intrinsic Bayes factor, can often be used as a default measure to circumvent this indeterminacy. On the other hand, if reasonable (possibly proper) called intrinsic priors are available, the intrinsic Bayes factor can be approximated by calculating the ordinary Bayes factor with intrinsic priors. Additionally, the concept of the integral prior, inspired by the generalized expected posterior prior, often serves to mitigate the uncertainty in traditional Bayes factors. Consequently, the Bayes factor derived from this approach can effectively approximate the conventional Bayes factor. In this article, we present default Bayesian procedures when testing the zero inflation parameter in a zero-inflated Poisson distribution. Approximation methods are used to derive intrinsic and integral priors for testing the zero inflation parameter. A Monte Carlo simulation study is carried out to demonstrate theoretical outcomes, and two real datasets are analyzed to support the results found in this paper.