Abstract
One of the tasks of the Bayesian consulting statistician is to elicit prior information from his client who may be unfamiliar with parametric statistical models. In some cases it may be more illuminating to base a prior distribution for parameter θ on the transformed version F(⋎/θ), where F is the data distribution function and v is a designated reference value, rather than on θ directly. This approach is outlined and explored in various directions to assess its implications. Some applications are given, including general linear regression and transformed linear models.
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References
Apostol, T. M. (1957). Mathematical Analysis, Addison-Wesley, London.
Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion), J. Roy. Statist. Soc. Ser. B, 41, 113–147.
Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations (with discussion), J. Roy. Statist. Soc. Ser. B, 26, 211–252.
Chaloner, K. M. and Duncan, G. T. (1983). Assessment of a Beta prior distribution: PM elicitation, The Statistician, 32, 174–180.
Dawid, A. P., Stone, M. and Zidek, J. V. (1973). Marginalization paradoxes in Bayesian and structural inference (with discussion), J. Roy. Statist. Soc. Ser. B, 35, 189–233.
Dickey, J. M. (1980) Beliefs about beliefs, a theory of stochastic assessments of subjective probabilities, Probability and the Weighing of Evidence, Griffin, London.
Good, I. J. (1950). Probability and the Weighing of Evidence, Griffin, London.
Hartigan, J. (1964). Invariant prior distributions, Ann. Math. Statist., 35, 836–845.
Hogarth, R. M. (1975). Cognitive processes and the assessment of subjective probability distributions (with discussion), J. Amer. Statist. Assoc., 70, 271–289.
Kadane, J. B. (1980). Predictive and structural methods of eliciting prior distributions, Bayesian Analysis in Econometrics and Statistics, North-Holland, Amsterdam.
Kadane, J. B., Dickey, J. M., Winkler, R. L., Smith, W. S. and Peters, S. C. (1980). Interactive elicitation of opinion for a normal linear model, J. Amer. Statist. Assoc., 75, 845–854.
Lindley, D. V. (1972). Bayesian Statistics, A Review, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania.
Lindley, D. V., Tversky, A. and Brown, R. V. (1979). On the reconciliation of probability assessments (with discussion), J. Roy. Statist. Soc. Ser. A, 142, 146–180.
Novick, M. R. (1969). Multiparameter Bayesian indifference procedures, J. Roy. Statist. Soc. Ser. B, 31, 29–64.
Pericchi, L. R. (1981). A Bayesian approach to transformations to normality, Biometrika, 68, 35–43.
Savage, L. J. (1971). Elicitation of personal probabilities and expectations, J. Amer. Statist. Assoc., 66, 783–801.
Sweeting, T. J. (1984). On the choice of prior distribution for the Box-Cox transformed linear model, Biometrika, 71, 127–134.
Tversky, A. (1974). Assessing uncertainty (with discussion), J. Roy. Statist. Soc. Ser. B, 36, 148–159.
Winkler, R. L. (1967). The assessment of prior distributions in Bayesian analysis, J. Amer. Statist. Assoc., 62, 776–800.
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Crowder, M. Bayesian priors based on a parameter transformation using the distribution function. Ann Inst Stat Math 44, 405–416 (1992). https://doi.org/10.1007/BF00050695
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DOI: https://doi.org/10.1007/BF00050695