Abstract
Existence conditions for posterior mean of Bayesian logistic regression depend on both chosen prior distributions and a likelihood function. In logistic regression, different patterns of data points can lead to finite maximum likelihood estimates (MLE) or infinite MLE of the regression coefficients. Albert and Anderson [On the existence of maximum likelihood estimates in logistic regression models, Biometrika 71 1984, 1, 1–10] gave definitions of different types of data points, which are complete separation, quasicomplete separation and overlap. Conditions for the existence of the MLE for logistic regression models were proposed under different types of data points. Based on these conditions, we propose the necessary and sufficient conditions for the existence of posterior mean under different choices of prior distributions. In this paper, a general wide class of priors, which are informative priors and non-informative priors having proper distributions and improper distributions, are considered for the existence of posterior mean. In addition, necessary and sufficient conditions for the existence of posterior mean for an individual coefficient is also proposed.
References
[1] A. Albert and J. A. Anderson, On the existence of maximum likelihood estimates in logistic regression models, Biometrika 71 (1984), no. 1, 1–10. 10.1093/biomet/71.1.1Search in Google Scholar
[2] J. H. Albert and S. Chib, Bayesian analysis of binary and polychotomous response data, J. Amer. Statist. Assoc. 88 (1993), no. 422, 669–679. 10.1080/01621459.1993.10476321Search in Google Scholar
[3] H. M. Choi and J. P. Hobert, The Polya-gamma Gibbs sampler for Bayesian logistic regression is uniformly ergodic, Electron. J. Stat. 7 (2013), 2054–2064. 10.1214/13-EJS837Search in Google Scholar
[4] A. Gelman, J. B. Carlin, H. S. Stern and D. B. Rubin, Bayesian Data Analysis, Chapman & Hall/CRC, London, 2004. 10.1201/9780429258480Search in Google Scholar
[5] A. Gelman, A. Jakulin, M. G. Pittau and Y.-S. Su, A weakly informative default prior distribution for logistic and other regression models, Ann. Appl. Stat. 2 (2008), no. 4, 1360–1383. 10.1214/08-AOAS191Search in Google Scholar
[6] J. Ghosh, Y. Li and R. Mitra, On the use of Cauchy prior distributions for Bayesian logistic regression, Bayesian Anal. 13 (2018), no. 2, 359–383. 10.1214/17-BA1051Search in Google Scholar
[7] G. Heinze, A comparative investigation of methods for logistic regression with separated or nearly separated data, Stat. Med. 25 (2006), no. 24, 4216–4226. 10.1002/sim.2687Search in Google Scholar PubMed
[8] N. G. Polson, J. G. Scott and J. Windle, Bayesian inference for logistic models using Pólya–Gamma latent variables, J. Amer. Statist. Assoc. 108 (2013), no. 504, 1339–1349. 10.1080/01621459.2013.829001Search in Google Scholar
[9] V. Roy and J. P. Hobert, Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression, J. R. Stat. Soc. Ser. B Stat. Methodol. 69 (2007), no. 4, 607–623. 10.1111/j.1467-9868.2007.00602.xSearch in Google Scholar
[10] P. L. Speckman, J. Lee and D. Sun, Existence of the MLE and propriety of posteriors for a general multinomial choice model, Statist. Sinica 19 (2009), no. 2, 731–748. Search in Google Scholar
[11] J. Wakefield, Bayesian and Frequentist Regression Methods, Springer, New York, 2013. 10.1007/978-1-4419-0925-1Search in Google Scholar
[12] R. W. M. Wedderburn, On the existence and uniqueness of the maximum likelihood estimates for certain generalized linear models, Biometrika 63 (1976), no. 1, 27–32. 10.1093/biomet/63.1.27Search in Google Scholar
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