Abstract
Posterior mode estimators are proposed, which arise from simply expressed prior opinion about expected outcomes, roughly as follows: a conjugate family of prior distributions is determined by a given variance function. Using a conjugate prior, a posterior mode estimator and its estimated (co-)variances are obtained through conventional maximum likelihood computations, by means of small alterations to the observed outcomes and/or to the modelled variance function. Within the conjugate family, for purposes of inference about the regression vector, a reference prior is proposed for a given choice of linear design of the canonical link. The resulting approximate reference inferences approximate the Bayesian inferences which arise from a “minimally informative” reference prior. A set of subjective prior upper and lower percentage points for the expected outcomes can be used to determine a conjugate family member. Alternatively, a set of subjective prior means and standard deviations determines a member. The subfamily of priors determinable by percentage points either includes or approximates the proposed reference prior.
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References
Aitkin, M., Anderson, D., Francis, B. and Hinde, J. (1989). Statistical Modelling in GLIM, Oxford, New York.
Albert, J. H. (1988). Computational methods using a Bayesian hierarchical generalized linear model. J. Amer. Statist. Assoc., 83, 1037–1044.
Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed., Springer, New York.
Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion), J. R. Statist. Soc. Ser. B, 41, 113–147.
BMDP Statistical Software (1983). University of California Press, Berkeley.
Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis, Addison-Wesley, Reading, Massachusetts.
Chang, T. and Eaves, D. M. (1990). Reference priors for the orbit in a group model, Ann. Math. Stud., 18, 1595–1614.
Cox, D. R. (1970). The Analysis of Binary Data, Chapman and Hall, London.
CYTEL Software Corp. (1990). Publicity for the package StatXact, based on work of Mehta, C. and Patel, N., Cambridge, Massachusetts.
Dobson, A. J. (1983). An Introduction to Statistical Modelling, Chapman and Hall, New York.
Goodman, L. (1970). The multivariate analysis of qualitative data: interactions among multiple classifications, J. Amer. Statist. Assoc., 65, 226–256.
Jennrich, R. I. and Moore, R. H. (1975). Maximum likelihood estimation by means of nonlinear least squares, BMDP Technical Report, #9, BMDP Statistical Software, Inc., Los Angeles, California.
Marcus, M. and Minc, H. (1966). Modern University Algebra, Macmillan, New York.
McCullagh, P. (1983). Quasi-likelihood functions, Ann. Statist., 11, 59–67.
McCullagh, P. and Nelder, J. (1983). Generalized Linear Models, Chapman and Hall, London.
Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized Linear Models. J. R. Statist. Soc. Ser. A, 135, 370–384.
Pearl, M. (1973). Matrix Theory and Finite Mathematics, McGraw-Hill, New York.
Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalised linear models, and the Gauss-Newton method, Biometrika, 61, 439–447.
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The research of the first-named author was funded in part by a Natural Sciences and Engineering Research Council grant.
The second named author gratefully acknowledges the support of the National Science Foundation, grant #DMS-8901494 and of the Kansas Geological Survey where he visited during the term of the majority of this research.
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Eaves, D.M., Chang, T. Posterior mode estimation for the generalized linear model. Ann Inst Stat Math 44, 417–434 (1992). https://doi.org/10.1007/BF00050696
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DOI: https://doi.org/10.1007/BF00050696