# zbMATH — the first resource for mathematics

Bayes spaces: use of improper distributions and exponential families. (English) Zbl 1280.62030
Summary: Bayes spaces are vector spaces of sigma-additive positive measures. Proportional measures are considered equivalent and can be represented by densities with respect to a fixed dominating measure. The addition in these spaces is perturbation. It corresponds to Bayes’ theorem, which appears as a linear operation. Bayes spaces, with continuous dominating measures, contain finite and infinite measures. Finite measures are equivalent to probability measures. Infinite measures include what in Bayesian statistics are called improper priors and non-integrable likelihood functions, justifying the use of such improper densities in Bayes’ theorem.
Many concepts of probability theory can be handled in a natural way in the context of Bayes spaces. Particularly, an exponential family of probability densities appears as a cone contained in an affine subspace of the Bayes space. The framework of Bayes spaces allows an easy handling of exponential families and their extensions to improper distributions. Furthermore, the vector space structure of Bayes spaces allows the definition of derivatives of densities. In Bayesian statistics, these derivatives are a new tool to examine sensitivity of posterior distributions with respect to both observed data and prior changes.

##### MSC:
 62F15 Bayesian inference 62A01 Foundations and philosophical topics in statistics
##### Keywords:
simplex; Aitchison geometry; derivatives; sensitivity
Full Text:
##### References:
 [1] Aitchison J.: The statistical analysis of compositional data (with discussion). J. R. Stat. Soc. Ser. B (Stat. Methodol.) 44(2), 139–177 (1982) · Zbl 0491.62017 [2] Aitchison J.: Principal component analysis of compositional data. Biometrika 70(1), 57–65 (1983) · Zbl 0515.62057 · doi:10.1093/biomet/70.1.57 [3] Aitchison, J.: The Statistical Analysis of Compositional Data. Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (reprinted in 2003 with additional material by The Blackburn Press) (1986) [4] Aitchison J.: On criteria for measures of compositional difference. Math. Geol. 24(4), 365–379 (1992) · Zbl 0970.86531 · doi:10.1007/BF00891269 [5] Aitchison, J., Barceló-Vidal, C., Egozcue, J.J., Pawlowsky-Glahn, V.: A concise guide for the algebraic-geometric structure of the simplex, the sample space for compositional data analysis. In: Bayer, U., Burger, H., Skala, W. (eds.) Proceedings of IAMG’02–The eigth annual conference of the International Association for Mathematical Geology, vol. I and II, pp. 387–392. Selbstverlag der Alfred-Wegener-Stiftung, Berlin (2002) [6] Aitchison J., Barceló-Vidal C., Martín-Fernández J., Pawlowsky-Glahn V.: Reply to Letter to the Editor by S. Rehder and U. Zier. Math. Geol. 33(7), 849–860 (2001) · Zbl 1101.86310 · doi:10.1023/A:1010954915624 [7] Azzalini A., Dalla Valle A.: The multivariate skew-normal distribution. Biometrika 83(4), 715–726 (1996) · Zbl 0885.62062 · doi:10.1093/biomet/83.4.715 [8] Barceló-Vidal, C., Martín-Fernández, J.A., Pawlowsky-Glahn, V.: Mathematical foundations of compositional data analysis. In: Ross, G. (ed.) Proceedings of IAMG’01–The Sixth Annual Conference of the International Association for Mathematical Geology, pp. 1–20 (2001) (CD-ROM) [9] Bernardo J.M.: Reference posterior distributions for Bayesian statistics. J. R. Stat. Soc. B 41(2), 113–147 (1979) · Zbl 0428.62004 [10] Bernardo, J.M.: Bayesian statistics. Encyclopedia of life support systems (EOLSS), a integrated virtual library. In: Viertl, R. (ed.) Probability and Statistics. UNESCO 2003, Oxford. http://www.eolss.net (2003) [11] Bernardo J.M., Ramón J.M.: An introduction to Bayesian reference anlysis: inference on the ratio of multinomial parameters. Statistician 47(Part 1), 101–145 (1998) [12] Bernardo J.M., Smith A.F.M.: Bayesian Theory. Wiley, Chichester (1986) [13] Billheimer D., Guttorp P., Fagan W.: Statistical interpretation of species composition. J. Am. Stat. Assoc. 96(456), 1205–1214 (2001) · Zbl 1073.62573 · doi:10.1198/016214501753381850 [14] Birnbaum A.: On the foundations of statistical inference (with discussion). J. Am. Stat. Assoc. 57, 269–326 (1962) · Zbl 0107.36505 · doi:10.1080/01621459.1962.10480660 [15] van den Boogaart K.G., Egozcue J.J., Pawlowsky-Glahn V.: Bayes linear spaces. Statistics and operations research transactions. SORT 34(2), 201–222 (2010) · Zbl 1208.62003 [16] Box G.E.P., Tiao G.C.: Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading (1973) · Zbl 0271.62044 [17] Egozcue, J.J., Barceló-Vidal, C., Martín-Fernández, J.A., Jarauta-Bragulat, E., Díaz-Barrero, J.L., Mateu-Figueras, G.: Elements of simplicial linear algebra and geometry. In: Pawlowsky-Glahn, V., Buccianti, A. (eds.) Compositional Data Analysis: Theory and Applications, Wiley, Chichester (2011) [18] Egozcue J.J., Díaz-Barrero J.L., Pawlowsky-Glahn V.: Hilbert space of probability density functions based on Aitchison geometry. Acta Mathematica Sinica (English Series) 22(4), 1175–1182 (2006) · Zbl 1113.46016 · doi:10.1007/s10114-005-0678-2 [19] Egozcue, J.J., Jarauta-Bragulat, E., Díaz-Barrero, J.L.: Calculus of simplex-valued functions. In: Pawlowsky-Glahn, V., Buccianti, A. (eds.), Compositional Data Analysis: Theory and Applications, Wiley, Chichester (2011) [20] Jeffreys, H.: Theory of probability, 3rd edn. Oxford University Press, London (first edition 1939) (1961) · Zbl 0116.34904 [21] Pawlowsky-Glahn V., Egozcue J.J.: Geometric approach to statistical analysis on the simplex. Stoch. Environ. Res. Risk Assess. (SERRA) 15(5), 384–398 (2001) · Zbl 0987.62001 · doi:10.1007/s004770100077 [22] Robert C.P.: The Bayesian Choice. A Decision-Theoretic Motivation. Springer, New York (1994) · Zbl 0808.62005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.