Intrinsic priors for comparing zero-inflation parameters in Poisson models

Abstract

Prior elicitation is an important issue in both objective and subjective Bayesian inferences. In hypothesis testing and model selection, choosing appropriate prior distributions becomes significantly more critical. In an objective Bayesian analysis, one utilizes noninformative priors such as Jeffreys priors or reference priors for hypothesis testing which are often improper, making unspecified constants to be contained in the Bayes factor. Thus, the resulting Bayes factor should be adjusted. In this paper, we consider default Bayes procedures for testing zero-inflation parameters in a zero-inflated Poisson distribution. In particular, we derive a set of intrinsic priors based on an approximation procedure. Extensive simulations and analyses of two real datasets are performed to support the methodology developed in the paper. It is shown that the proposed Bayesian and frequentist approaches yield similar comparable results.

Keywords

• Fractional Bayes factor
• intrinsic Bayes factor
• intrinsic prior
• training sample

Supporting Institution

National Research Foundation of Korea

Project Number

NRF-2021R1A2C1005271; NRF-2020R1A2C3A01003550

Thanks

Y. Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2C3A01003550). S. W. Kim’s research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1A2C1005271).

References

1. [1] I. A. Almod´ovar-Rivera and L. R. Pericchi-Guerra, An objective and robust Bayes factor for the hypothesis test one sample and two population means, Entropy 26 (1), 1-25, 2024.
2. [2] A. M. Azexedo, ´I. J. Silva, M. C. Nery, H. P. Rocha and R. A. Santana, Counting models for overdispersed data: A review with application to tuberculosis data, Braz. J. Biometrics 41 (3), 274-286, 2023.
3. [3] M. J. Bayarri, J. O. Berger and G. S. Datta, Objective Bayes testing of Poisson versus inflated Poisson models, IMS Collect. 3, 105-121, 2008.
4. [4] J. O. Berger, The case for objective Bayesian analysis, Bayesian Anal. 1 (3), 385-402, 2006.
5. [5] J. O. Berger and J. M. Bernardo, Estimating a product of means: Bayesian analysis with reference priors, J. Am. Stat. Assoc. 84 (405), 200-207, 1989.
6. [6] J. O. Berger and J. Moreta, Default Bayes factors for nonnested hypothesis testing, J. Am. Stat. Assoc. 94 (446), 542-554, 1999.
7. [7] J. O. Berger and L. Pericchi, The intrinsic Bayes factor for model selection and prediction, J. Am. Stat. Assoc. 91 (433), 109-122, 1996.
8. [8] S. Chen, Y. Li, J. Kim and S. W. Kim, Bayesian change point analysis for extreme daily precipitation, Int. J. Climatol. 37 (7), 3123-3137, 2017.

9. [9] R. Clare, A universal robust bound for the intrinsic Bayes factor, Ph.D. dissertation, Univ. Puerto Rico, 2024.
10. [10] A. C. Cohen, Estimation in mixtures of discrete distributions, in Proc. Int. Symp. Discrete Distrib., Montreal, 373-378, 1963.
11. [11] C. Conigliani and A. O’Hagan, Sensitivity of the fractional Bayes factor to prior distributions, Can. J. Stat. 28 (2), 343-352, 2000.
12. [12] G. Consonni, D. Fouskakis, B. Liseo and I. Ntzoufras, Prior distributions for objective Bayesian analysis, Bayesian Anal. 13 (2), 627-679, 2018.
13. [13] X. Gu, J. Mulder and H. Hoijtink, Approximated adjusted fractional Bayes factors: A general method for testing informative hypotheses, Br. J. Math. Stat. Psychol. 71 (2), 229-261, 2018.
14. [14] D. B. Hall, Zero-inflated Poisson and binomial regression with random effects: A case study, Biometrics 56 (4), 1030-1039, 2000.
15. [15] Y. Han, H. Hwang, H. K. T. Ng and S. W. Kim, Default Bayesian testing for the Zero-inflated Poisson distribution, Stat. Interface 17 (4), 623-634, 2024.
16. [16] H. Jeffreys, Theory of Probability, 3rd ed., Oxford Univ. Press, 1961.
17. [17] R. E. Kass and A. E. Raftery, Bayes factors, J. Am. Stat. Assoc. 90 (430), 773-795, 1995.
18. [18] S. W. Kim, Intrinsic priors for testing exponential means, Stat. Probab. Lett. 46 (2), 195-201, 2000.
19. [19] S. W. Kim and D. Kim, Intrinsic priors for two-sample tests in normal populations, Commun. Stat.-Theory Methods 31 (7), 1091-1105, 2002.
20. [20] S. W. Kim and D. Sun, Intrinsic priors for model selection using an encompassing model with applications to censored failure time data, Lifetime Data Anal. 6, 251-269, 2000.
21. [21] D. Lambert, Zero-inflated Poisson regression, with an application to defects in manufacturing, Technometrics 34 (1), 1-14, 1992.
22. [22] K. Lee, Y. Joo, J. J. Song and D. W. Harper, Analysis of zero-inflated clustered count data: A marginalized model approach, Comput. Stat. Data Anal. 55 (1), 824-837, 2011.
23. [23] H. K. Lim, W. K. Li and P. L. H. Yu, Zero-inflated Poisson regression mixture model, Comput. Stat. Data Anal. 71, 151-158, 2014.
24. [24] D. L. Long, J. S. Preisser, A. H. Herring and C. E. Golin, A marginalized zero-inflated Poisson regression model with random effects, J. R. Stat. Soc. C 64 (5), 815-830, 2015.
25. [25] K. Mahmood and F. Havva, Inferences for the inflation parameter in the zip distributions: The method of moments, Stat. Methodol. 8 (4), 377-388, 2011.
26. [26] Y. Min and A. Agresti, Random effect models for repeated measures of zero-inflated count data, Stat. Modell. 5 (1), 1-19, 2005.
27. [27] E. Moreno, Objective Bayesian methods for one-sided testing, Test 14 (1), 181-198, 2005.
28. [28] J. Mullahy, Specification and testing of some modified count data models, J. Econom. 33 (3), 341-365, 1986.
29. [29] B. Neelon and D. Chung, The LZIP: A Bayesian latent factor model for correlated zero-inflated counts, Biometrics 73 (1), 185-196, 2017.
30. [30] A. O’Hagan, Fractional Bayes factors for model comparison, J. R. Stat. Soc. B 57 (1), 99-118, 1995.
31. [31] L. Perreault, J. Bernier, B. Bobée and E. Parent, Bayesian change-point analysis in hydrometeorological time series. Part 2. Comparison of change-point models and forecasting, J. Hydrol. 235 (3-4), 242-263, 2000.
32. [32] J. Schwartz and D. Giles, Bias-reduced maximum likelihood estimation of the zeroinflated Poisson distribution, Commun. Stat.-Theory Methods 45 (2), 465-478, 2016.
33. [33] S. Sivaganesan and D. Jiang, Objective Bayesian testing of a Poisson mean, Commun. Stat.-Theory Methods 39 (11), 1887-1897, 2010.
34. [34] Y. Wang and L. Pericchi, A bridge between cross-validation Bayes factors and geometric intrinsic Bayes factors, arXiv: 2006.06495v1.
35. [35] X. Xiao, Y. Tang, A. Xu and G. Wang, Bayesian inference for zero-and-one-inflated geometric distribution regression model using Polya-gamma latent variables, Commun. Stat.-Theory Methods 49 (15), 3730-3743, 2020.
36. [36] H. Xu, M. Xie and T. N. Goh, Objective Bayes analysis of zero-inflated Poisson distribution with application to healthcare data, IIE Trans. 46 (8), 843-852, 2014.
37. [37] K. K. Yau and A. H. Lee, Zero-inflated Poisson regression with random effects to evaluate an occupational injury prevention programme, Stat. Med. 20 (19), 2907-2920, 2001.