Abstract
This paper investigates the existence of Bayesian estimates for polychotomous quantal response models using a uniform improper prior distribution on the regression parameters. Necessary and sufficient conditions for the propriety of the posterior distribution with a general link function are established. In addition, the sufficient conditions for the existence of the posterior moments and the posterior moment generating function are obtained. It is also found that the propriety guarantees the existence of the maximum likelihood estimate.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Agresti, A. (1990). Categorical Data Analysis, Wiley, New York.
Box, G. E. P. and Tiao, G. C. (1992). Bayesian Inference in Statistical Analysis, Wiley, New York.
Chen, M.-H. (1994). Importance weighted marginal Bayesian posterior density estimation, J. Amer. Statist. Assoc., 89, 818–824.
Chen, M.-H. and Dey, D. K. (1998). Bayesian modeling of correlated binary responses via scale mixture of multivariate normal link functions, Sankhya, Ser. A, 60, 322–343.
Chen, M.-H. and Shao, Q.-M. (1999a). Propriety of posterior distribution for dichtomous quantal response models with general link functions, Proceedings of the American Mathematical Society (to appear).
Chen, M.-H. and Shao, Q.-M. (1999b). Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Statist., 8, 69–92.
CPLEX Optimization (1992). CPLEX, CPLEX Optimization Inc, Incline Village, Nevada.
Gelland, A. E. and Smith, A. F. M. (1990). Sampling based approaches to calculating marginal densities, J. Amer. Statist. Assoc., 85, 398–409.
Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
Gilks, W. R. and Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling, Applied Statistics, 41, 337–348.
Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 97–109.
Ibrahim, J. G. and Laud, P. W. (1991). On Bayesian analysis of generalized linear models using Jeffrey's prior, J. Amer. Statist. Assoc., 86, 981–986.
Jaynes, E. T. (1982). On the rationale of maximum-entropy methods, Proceedings of IEEE, 70, 939–952.
McCullagh, P. and Nekler, J. A. FRS (1989). Generalized Linear Models, 2nd ed., Chapman & Hall, London.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines, Journal of Chemical Physics, 21, 1087–1092.
Rossi, P. (1996). Existence of Bayes estimators for the binomial logit model, Bayesian Analysis in Statistics and Econometrics--Essays in Honor of Arnold Zellner (eds. D. A. Berry, K. M. Chaloner and J. K. Geweke), 91–100, Wiley, New York.
Stokes, M. E., Davis, C. S. and Koch, G. G. (1995). Categorial Data Analysis Using the SAS System, SAS Institute Inc., Cary, North Carolina.
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion), Ann. Statist., 22, 1701–1762.
Wedderburn, R. W. M. (1976). On the existence and uniqueness of the maximum likelihood estimates for certain generalized linear models, Biometrika, 63, 27–32.
Zellner, A. (1997). Past and recent results on maximal data information priors, Bayesian Analysis in Econometrics and Statistics: The Zellner View and Papers, 127–148, Edward Elgar, Cheltenham.
Zellner, A. and Rossi, P. E. (1984). Bayesian analysis of dichotomous quantal response models, J. Econometrics, 25, 365–393.
Author information
Authors and Affiliations
About this article
Cite this article
Chen, MH., Shao, QM. Existence of Bayesian Estimates for the Polychotomous Quantal Response Models. Annals of the Institute of Statistical Mathematics 51, 637–656 (1999). https://doi.org/10.1023/A:1004027028943
Issue Date:
DOI: https://doi.org/10.1023/A:1004027028943