Is there a formal explanation of the concept of "improper prior" in Bayesian statistics?

Question

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?

Answer 1

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.

1. Bernardo, J. M. (1979), Reference posterior distributions for Bayesian inferences (with discussion). Journal of the Royal Statistical Society, Series B,41, 113-147.
2. Hartigan, J. A. (1964), Invariant prior distributions. The Annals of Mathematical Statistics, 35, 836-845.
3. Jeffreys, H. (1961), Theory of Probability. Third Edition. London: Oxford University Press.
4. Villegas, C.
   1. (1971), On Haar priors. In: Foundations of Statistical Inference, edited by V. P. Godambe and D. A. Sproot. Toronto: Holt, Rinehart, and Winston.
   2. (1972), Bayesian inference in linear relations. Annals of Mathematical Statistics, 43, 1767-1791.
   3. (1977a), Inner statistical inference. Journal of the American Statistical Association, 72, 453-458.
   4. (1977b), On the representation of ignorance. Journal of the American Statistical Association, 72, 653-654.
5. 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.

1. 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.
2. 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.

1. DeFinetti, B. (1974), Theory of Probability, Vols. I and II. New York: Wiley.
2. Hartigan, J. (1983), Bayes Theory. New York: Springer-Verlag.

He also mentions the following works in the same context.

1. Berti, P., Regazzini, E. and Rigo, P. (1991), Coherent statistical inference and Bayes theorem. Annals of Statistics, 19, 366-381.
2. Heath, D. and Sudderth, W. D. (1989), Coherent inference from improper priors and from finitely additive priors. Annals of Statistics, 17, 907-919.
3. 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.
4. 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.
5. Stone, M. (1976), Strong inconsistency from uniform priors. Journal of the American Statistical Association, 71, 114-125.
6. 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.