A general framework for updating belief distributions
We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special ca...
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| Vydané v: | Journal of the Royal Statistical Society. Series B, Statistical methodology Ročník 78; číslo 5; s. 1103 - 1130 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
England
Blackwell Publishing Ltd
01.11.2016
John Wiley & Sons Ltd Oxford University Press John Wiley and Sons Inc |
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| ISSN: | 1369-7412, 1467-9868 |
| On-line prístup: | Získať plný text |
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| Abstract | We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted. |
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| AbstractList | We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted. Summary We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data‐generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted. We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted. |
| Author | Holmes, C. C. Walker, S. G. Bissiri, P. G. |
| AuthorAffiliation | 1 University of Milano‐Bicocca Italy 2 University of Oxford UK 3 University of Texas at Austin USA |
| AuthorAffiliation_xml | – name: 1 University of Milano‐Bicocca Italy – name: 2 University of Oxford UK – name: 3 University of Texas at Austin USA |
| Author_xml | – sequence: 1 givenname: P. G. surname: Bissiri fullname: Bissiri, P. G. organization: University of Milano-Bicocca, Italy – sequence: 2 givenname: C. C. surname: Holmes fullname: Holmes, C. C. email: c.holmes@stats.ox.ax.uk, c.holmes@stats.ox.ax.uk organization: University of Oxford, UK – sequence: 3 givenname: S. G. surname: Walker fullname: Walker, S. G. organization: University of Texas at Austin, USA |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/27840585$$D View this record in MEDLINE/PubMed |
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| Keywords | Loss function Self‐information loss function Decision theory Provably approximately correct Bayes methods General Bayesian updating Gibbs posteriors Information Maximum entropy Generalized estimating equations |
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| PublicationCentury | 2000 |
| PublicationDate | November 2016 |
| PublicationDateYYYYMMDD | 2016-11-01 |
| PublicationDate_xml | – month: 11 year: 2016 text: November 2016 |
| PublicationDecade | 2010 |
| PublicationPlace | England |
| PublicationPlace_xml | – name: England – name: Oxford – name: Hoboken |
| PublicationTitle | Journal of the Royal Statistical Society. Series B, Statistical methodology |
| PublicationTitleAlternate | J. R. Stat. Soc. B |
| PublicationYear | 2016 |
| Publisher | Blackwell Publishing Ltd John Wiley & Sons Ltd Oxford University Press John Wiley and Sons Inc |
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Zhang, T. (2006b) Information theoretical upper and lower bounds for statistical estimation. IEEE Trans. Inform. Theor., 52, 1307-1321. 2010; 11 2002; 18 1997; 46 1982; 11 2008; 36 2005; 20 1951; 22 2010; 140 2008; 1 1965; 19 1981; 43 1998; 44 2005; 25 1966; 28 2006b; 52 2001 2000 2000; 15 2005; 102 1997; 16 1964; 35 1999; 50 1988; 42 1937; 7 1998; 54 2012; 22 1979; 7 1944 2012 2002; 30 1999; 27 2008; 17 1982; 77 2009 1998 1954 1997 2007 2006 1994 1993 2003 1992 2013; 143 1996; 91 1972; 67 2006a; 34 1959 2012b; 64 2001; 63 1999 2005; 6 2012a; 142 1972; 34 2003; 65 Zellner (2023021709301395800_rssb12158-cit-0057) 1988; 42 Goldstein (2023021709301395800_rssb12158-cit-0023) 1981; 43 Royall (2023021709301395800_rssb12158-cit-0049) 2003; 65 Hüber (2023021709301395800_rssb12158-cit-0029) 1964; 35 Kass (2023021709301395800_rssb12158-cit-0038) 1996; 91 Hüber (2023021709301395800_rssb12158-cit-0030) 2009 Catoni (2023021709301395800_rssb12158-cit-0010) 2003 de Finetti (2023021709301395800_rssb12158-cit-0019) 1937; 7 Savage (2023021709301395800_rssb12158-cit-0050) 1954 Freund (2023021709301395800_rssb12158-cit-0020) 1965; 19 Tibshirani (2023021709301395800_rssb12158-cit-0054) 1997; 16 Merhav (2023021709301395800_rssb12158-cit-0045) 1998; 44 Jasra (2023021709301395800_rssb12158-cit-0036) 2005; 20 Gardner (2023021709301395800_rssb12158-cit-0021) 1959 Hartigan (2023021709301395800_rssb12158-cit-00501) 1972; 67 Cheng (2023021709301395800_rssb12158-cit-0012) 2000 Cesa-Bianchi (2023021709301395800_rssb12158-cit-0011) 2006 Cox (2023021709301395800_rssb12158-cit-0013) 1972; 34 Berger (2023021709301395800_rssb12158-cit-0004) 1993 Tanay (2023021709301395800_rssb12158-cit-0053) 2002; 18 Shawe-Taylor (2023021709301395800_rssb12158-cit-0052) 1997 Jiang (2023021709301395800_rssb12158-cit-0037) 2008; 36 Cooley (2023021709301395800_rssb12158-cit-0048) 2009 Zhang (2023021709301395800_rssb12158-cit-0059) 2006; 52 Bar-Hillel (2023021709301395800_rssb12158-cit-0003) 1982; 11 Bernardo (2023021709301395800_rssb12158-cit-0005) 1979; 7 Ali (2023021709301395800_rssb12158-cit-0001) 1966; 28 Walker (2023021709301395800_rssb12158-cit-0056) 2001; 63 Hutchison (2023021709301395800_rssb12158-cit-0031) 1999; 50 Kullback (2023021709301395800_rssb12158-cit-0040) 1951; 22 Müller (2023021709301395800_rssb12158-cit-0046) 2012 Bissiri (2023021709301395800_rssb12158-cit-0007) 2010; 140 Ibrahim (2023021709301395800_rssb12158-cit-0034) 1999; 27 McAllester (2023021709301395800_rssb12158-cit-0044) 1998 Ibrahim (2023021709301395800_rssb12158-cit-0035) 2001 Hirshleifer (2023021709301395800_rssb12158-cit-0027) 1992 Alquier (2023021709301395800_rssb12158-cit-0002) 2008; 17 Diaconis (2023021709301395800_rssb12158-cit-0015) 1982; 77 Heard (2023021709301395800_rssb12158-cit-0026) 2005; 102 Bissiri (2023021709301395800_rssb12158-cit-0009) 2012; 64 Faraggi (2023021709301395800_rssb12158-cit-0018) 1998; 54 Seldin (2023021709301395800_rssb12158-cit-0051) 2010; 11 Goldstein (2023021709301395800_rssb12158-cit-0024) 2007 Fan (2023021709301395800_rssb12158-cit-0017) 2002; 30 Volinsky (2023021709301395800_rssb12158-cit-0055) 1997; 46 Langford (2023021709301395800_rssb12158-cit-0041) 2005; 6 Doucet (2023021709301395800_rssb12158-cit-0016) 2012 Bissiri (2023021709301395800_rssb12158-cit-0008) 2012; 142 von Neumann (2023021709301395800_rssb12158-cit-0047) 1944 Hoff (2023021709301395800_rssb12158-cit-0028) 2013; 143 Ibrahim (2023021709301395800_rssb12158-cit-0033) 2000; 15 Marin (2023021709301395800_rssb12158-cit-0043) 2012; 22 Bernardo (2023021709301395800_rssb12158-cit-0006) 1994 Hastie (2023021709301395800_rssb12158-cit-0025) 2009 Key (2023021709301395800_rssb12158-cit-0039) 1999 Zhang (2023021709301395800_rssb12158-cit-0058) 2006; 34 Hutchison (2023021709301395800_rssb12158-cit-0032) 2008; 1 Datta (2023021709301395800_rssb12158-cit-0014) 2005; 25 |
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| SubjectTerms | Bayesian analysis Bayesian theory Beliefs Coherence Data Decision theory Density equations Frame analysis General Bayesian updating Generalized estimating equations Gibbs posteriors Inference Information Joints Learning Loss function Mathematical models Maximum entropy Original Parameters probability Provably approximately correct Bayes methods Self-information loss function Statistics Studies Subjectivity |
| Title | A general framework for updating belief distributions |
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