Abstract
An alternative notion of conditional probability (say AN) is discussed and investigated. If compared with the usual notion (regular conditional distributions), AN gives up the measurability constraint but requires a properness condition. An existence result for AN is provided. Also, some consequences of AN are pointed out, with reference to Bayesian statistics, exchangeability and compatibility.
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References
Berti, P., Rigo, P.: On the existence of inferences which are consistent with a given model. Ann. Stat. 24, 1235–1249 (1996)
Berti, P., Rigo, P.: Sufficient conditions for the existence of disintegrations. J. Theor. Probab. 12, 75–86 (1999)
Berti, P., Rigo, P.: On coherent conditional probabilities and disintegrations. Ann. Math. Artif. Intell. 35, 71–82 (2002)
Berti, P., Rigo, P.: 0–1 laws for regular conditional distributions. Ann. Probab. 35, 649–662 (2007)
Berti, P., Rigo, P.: A conditional 0–1 law for the symmetric sigma-field. J. Theor. Probab. 21, 517–526 (2008)
Berti, P., Dreassi, E., Rigo, P.: Compatibility results for conditional distributions. J. Multivar. Anal. 125, 190–203 (2014)
Berti, P., Rigo, P.: Finitely additive mixtures of probability measures, submitted, (2018) currently available at: http://www-dimat.unipv.it/~rigo/bprv.pdf
Blackwell, D., Dubins, L.E.: On existence and non-existence of proper, regular, conditional distributions. Ann. Probab. 3, 741–752 (1975)
Dubins, L.E.: Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Probab. 3, 88–99 (1975)
Dubins, L.E., Prikry, K.: On the existence of disintegrations. In: Azema, J., Emery, M., Meyer, P.-A., Yor, M (eds.) Seminaire de Probab. XXIX, Lect. Notes in Math., Springer, 1613, 248–259 (1995)
Heath, D., Sudderth, W.D.: On finitely additive priors, coherence and extended admissibility. Ann. Stat. 6, 333–345 (1978)
Heath, D., Sudderth, W.D.: Coherent inference from improper priors and from finitely additive priors. Ann. Stat. 17, 907–919 (1989)
Jirina, M.: Conditional probabilities on \(\sigma \)-algebras with countable basis. Czech. Math. J. 4(79), 372–380. English translation in: Selected Transl. Math. Stat. and Prob., Am. Math. Soc., 2, 79–86 (1962) (1954)
Lane, D.A., Sudderth, W.D.: Coherent and continuous inference. Ann. Stat. 11, 114–120 (1983)
Maitra, A., Ramakrishnan, S.: Factorization of measures and normal conditional distributions. Proc. Am. Math. Soc. 103, 1259–1267 (1988)
Prikry, K., Sudderth, W.D.: Singularity with respect to strategic measures, Illinois. J. Math. 43, 139–153 (1982)
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Berti, P., Dreassi, E. & Rigo, P. A notion of conditional probability and some of its consequences. Decisions Econ Finan 43, 3–15 (2020). https://doi.org/10.1007/s10203-019-00256-9
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DOI: https://doi.org/10.1007/s10203-019-00256-9
Keywords
- Bayesian inference
- Conditional probability
- Disintegrability
- Regular conditional distribution
- Strategy
- Sufficiency