Conflations of probability distributions

Author:
Theodore P. Hill

Journal:
Trans. Amer. Math. Soc. **363** (2011), 3351-3372

MSC (2000):
Primary 60E05; Secondary 62B10, 94A17

DOI:
https://doi.org/10.1090/S0002-9947-2011-05340-7

Published electronically:
January 5, 2011

MathSciNet review:
2775811

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Abstract | References | Similar Articles | Additional Information

Abstract: The *conflation* of a finite number of probability distributions is a consolidation of those distributions into a single probability distribution , where intuitively is the conditional distribution of independent random variables with distributions , respectively, given that . Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. is shown to be the unique probability distribution that minimizes the loss of Shannon information in consolidating the combined information from into a single distribution , and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. In that sense, conflation may be viewed as an optimal method for combining the results from several different independent experiments. When are Gaussian, is Gaussian with mean the classical weighted-mean-squares reciprocal of variances. A version of the classical convolution theorem holds for conflations of a large class of a.c. measures.

**1.**J. Aitchison,*The statistical analysis of compositional data*, J. Roy. Statist. Soc. Ser. B**44**(1982), no. 2, 139–177. With discussion. MR**676206****2.**Aitken, A. (1934) On least-squares and linear combinations of observations,*Proc. Royal Soc. Edinburgh***55**, 42-48.**3.**Bracewell, R. (1999)*The Fourier Transform and its Applications*, 3rd Ed., McGraw-Hill.**4.**Kai Lai Chung,*A course in probability theory*, 3rd ed., Academic Press, Inc., San Diego, CA, 2001. MR**1796326****5.**J. J. Egozcue, J. L. Díaz-Barrero, and V. Pawlowsky-Glahn,*Hilbert space of probability density functions based on Aitchison geometry*, Acta Math. Sin. (Engl. Ser.)**22**(2006), no. 4, 1175–1182. MR**2245249**, https://doi.org/10.1007/s10114-005-0678-2**6.**John Elton and Theodore P. Hill,*A generalization of Lyapounov’s convexity theorem to measures with atoms*, Proc. Amer. Math. Soc.**99**(1987), no. 2, 297–304. MR**870789**, https://doi.org/10.1090/S0002-9939-1987-0870789-X**7.**Christian Genest and James V. Zidek,*Combining probability distributions: a critique and an annotated bibliography*, Statist. Sci.**1**(1986), no. 1, 114–148. With comments, and a rejoinder by the authors. MR**833278****8.**Lubinski, P. (2004) Averaging spectral shapes,*Monthly Notices of the Royal Astronomical Society***350**, 596-608.**9.**A. Liapounoff,*Sur les fonctions-vecteurs complètement additives*, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]**4**(1940), 465–478 (Russian, with French summary). MR**0004080****10.**`http://acwi.gov/methods/about/background.html`**11.**Mohr, P. and Taylor, B. (2000) CODATA recommended values of the fundamental physical constants: 1998,*Rev. Mod. Physics***72**, 351-495.**12.**Mohr, P., Taylor, B. and Newell, D. (2007) The fundamental physical constants,*Physics Today*, 52-55.**13.**Mohr, P., Taylor, B. and Newell, D. (2008) Recommended values of the fundamental physical constants: 2006,*Rev. Mod. Phys.***80**, 633-730.**14.**Alvin C. Rencher and G. Bruce Schaalje,*Linear models in statistics*, 2nd ed., Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2008. MR**2401650**

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Additional Information

**Theodore P. Hill**

Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Email:
hill@math.gatech.edu

DOI:
https://doi.org/10.1090/S0002-9947-2011-05340-7

Keywords:
Conflation of probability distributions,
Shannon information,
minimax likelihood ratio,
proportional consolidation,
product of probability density functions,
product of probability mass functions,
convolution theorem,
Gauss-Markov theorem,
best linear unbiased estimator,
maximum likelihood estimator.

Received by editor(s):
May 22, 2009

Received by editor(s) in revised form:
February 26, 2010

Published electronically:
January 5, 2011

Additional Notes:
This work was partially supported by the Netherlands Organization for Scientific Research (NWO)

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.