Is there a formal explanation of the concept of "improper prior" in Bayesian statistics? The Bayesian concept of "improper prior" seems to be surrounded with magic. Even formal, Bayesian-oriented books, such as Schervish's "Theory of Statistics", treat it with the heuristic hand waving ambiguity usual in less rigorous textbooks. Is there a book or article that deals with this concept/technique rigorously? Schervish mentions a couple formal attempts at tackling this concept, but also notes that they are radical in that they go beyond the standard axiomatization of probability theory, and hence open a whole "can of worms" (in his words). However, Schervish's book was published almost 20 years ago. Perhaps some advances in the field have been achieved in the meanwhile?
 A: 
In section 1.2.6, A Remark Regarding So-called "Improper" Prior Distributions of their text Elements of Bayesian Analysis. Marcel Dekker. 1990, Florens, Mouchart and Rolin cite the following references as a sample of works where more detailed analyses of "improper" or "noninformative" prior distributions may be found.



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Bernardo, J. M. (1979), Reference posterior distributions for Bayesian inferences (with discussion). Journal of the Royal Statistical Society, Series B,41, 113-147.


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Hartigan, J. A. (1964), Invariant prior distributions. The Annals of Mathematical Statistics, 35, 836-845.


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Jeffreys, H. (1961), Theory of Probability. Third Edition. London: Oxford University Press.


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Villegas, C.

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(1971), On Haar priors. In: Foundations of Statistical Inference, edited by V. P. Godambe and D. A. Sproot. Toronto: Holt, Rinehart, and Winston.


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(1972), Bayesian inference in linear relations. Annals of Mathematical Statistics, 43, 1767-1791.


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(1977a), Inner statistical inference. Journal of the American Statistical Association, 72, 453-458.


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(1977b), On the representation of ignorance. Journal of the American Statistical Association, 72, 653-654.




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Zellner. A. (1971), An Introduction to Bayesian Inference in Econometrics. New York: John Wiley.




In the first paragraph of said section, the following two works are mentioned.



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Dawid, A. P., Stone, M. and Zidek, J. V. (1973), Marginalization paradoxes in Bayesian and structural inference. Hournal of The Royal Statistical Society, Series B, 35 189-233.


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Mouchart, M. (1976), A note on Bayes theorem. Statistica, 36(2), 349-357.




Schervish (Theory of Statistics. Springer. 1995 (1st printing)) cites the following works (pp. 20-21) as expositing the two existing approaches to improper priors.



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DeFinetti, B. (1974), Theory of Probability, Vols. I and II. New York: Wiley.


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Hartigan, J. (1983), Bayes Theory. New York: Springer-Verlag.




He also mentions the following works in the same context.



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Berti, P., Regazzini, E. and Rigo, P. (1991), Coherent statistical inference and Bayes theorem. Annals of Statistics, 19, 366-381.


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Heath, D. and Sudderth, W. D. (1989), Coherent inference from improper priors and from finitely additive priors. Annals of Statistics, 17, 907-919.


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Kadane, J. B., Schervish, M. J. and Seidenfeld, T. (1985), Statistical implications of finitely additive probability. In P. Goel and A. Zellner (Eds.), Bayesian Inference and Decision Techniques with Applications: Essays in Honor of Bruno DeFinetti (pp. 59-76). Amsterdam: Elsevier Science Publishers.


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Schervish, M. J., Seidenfeld, T. and Kadane, J. B. (1984), The extent of non-conglomerability of finitely additive probabilities. Zeitschrift fuer Wahrscheinlichkeitstheorie, 66, 205-226.


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Stone, M. (1976), Strong inconsistency from uniform priors. Journal of the American Statistical Association, 71, 114-125.


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Stone, M. and Dawid, A. P. (1972), Un-Bayesian implications of improper Bayes inference in routine statistical problems. Biometrika, 59, 369-375.




Schervish also writes (ibid, p. 21):


An alternative to using improper priors is to do a robust Bayesian analysis


This "robust Bayesian analysis" is described in section 8.6.3 of Schervish's book.

