THE BERNSTEIN-VON MISES THEOREM AND NONREGULAR MODELS
We study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the "true" solution occurs on the boundary of the parameter space. We show that in this case Bayesian inference is consistent, and that the posterior distribution has not only Gauss...
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| Veröffentlicht in: | The Annals of statistics 2014-10, Vol.42 (5), p.1850-1878 |
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| container_end_page | 1878 |
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| container_issue | 5 |
| container_start_page | 1850 |
| container_title | The Annals of statistics |
| container_volume | 42 |
| creator | Bochkina, Natalia A. Green, Peter J. |
| description | We study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the "true" solution occurs on the boundary of the parameter space. We show that in this case Bayesian inference is consistent, and that the posterior distribution has not only Gaussian components as in the case of regular models (the Bernstein-von Mises theorem) but also has Gamma distribution components whose form depends on the behaviour of the prior distribution near the boundary and have a faster rate of convergence. We also demonstrate a remarkable property of Bayesian inference, that for some models, there appears to be no bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space. We illustrate the results on a problem from emission tomography. |
| doi_str_mv | 10.1214/14-AOS1239 |
| format | Article |
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We show that in this case Bayesian inference is consistent, and that the posterior distribution has not only Gaussian components as in the case of regular models (the Bernstein-von Mises theorem) but also has Gamma distribution components whose form depends on the behaviour of the prior distribution near the boundary and have a faster rate of convergence. We also demonstrate a remarkable property of Bayesian inference, that for some models, there appears to be no bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space. We illustrate the results on a problem from emission tomography.</description><identifier>ISSN: 0090-5364</identifier><identifier>EISSN: 2168-8966</identifier><identifier>DOI: 10.1214/14-AOS1239</identifier><language>eng</language><publisher>Hayward: Institute of Mathematical Statistics</publisher><subject>62F12 ; 62F15 ; Approximate posterior ; Approximation ; Asymptotic methods ; Bayesian analysis ; Bayesian inference ; Bernstein–von Mises theorem ; boundary ; Coordinate systems ; Inference ; Linear models ; Mathematical independent variables ; Mathematical models ; Matrices ; nonregular ; Parametric models ; Photons ; Pixels ; posterior concentration ; Probability distribution ; SPECT ; Statistical inference ; Studies ; Tomography ; total variation distance ; variance estimation in mixed models</subject><ispartof>The Annals of statistics, 2014-10, Vol.42 (5), p.1850-1878</ispartof><rights>Copyright © 2014 Institute of Mathematical Statistics</rights><rights>Copyright Institute of Mathematical Statistics Oct 2014</rights><rights>Copyright 2014 Institute of Mathematical Statistics</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c415t-6dc5ea4d0fb77363b7e9ef500fd3334717d05cd5d6845ca3dffe3e835d59bf9e3</citedby><cites>FETCH-LOGICAL-c415t-6dc5ea4d0fb77363b7e9ef500fd3334717d05cd5d6845ca3dffe3e835d59bf9e3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/43556346$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/43556346$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,776,780,799,828,881,921,27901,27902,57992,57996,58225,58229</link.rule.ids></links><search><creatorcontrib>Bochkina, Natalia A.</creatorcontrib><creatorcontrib>Green, Peter J.</creatorcontrib><title>THE BERNSTEIN-VON MISES THEOREM AND NONREGULAR MODELS</title><title>The Annals of statistics</title><description>We study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the "true" solution occurs on the boundary of the parameter space. We show that in this case Bayesian inference is consistent, and that the posterior distribution has not only Gaussian components as in the case of regular models (the Bernstein-von Mises theorem) but also has Gamma distribution components whose form depends on the behaviour of the prior distribution near the boundary and have a faster rate of convergence. We also demonstrate a remarkable property of Bayesian inference, that for some models, there appears to be no bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space. We illustrate the results on a problem from emission tomography.</description><subject>62F12</subject><subject>62F15</subject><subject>Approximate posterior</subject><subject>Approximation</subject><subject>Asymptotic methods</subject><subject>Bayesian analysis</subject><subject>Bayesian inference</subject><subject>Bernstein–von Mises theorem</subject><subject>boundary</subject><subject>Coordinate systems</subject><subject>Inference</subject><subject>Linear models</subject><subject>Mathematical independent variables</subject><subject>Mathematical models</subject><subject>Matrices</subject><subject>nonregular</subject><subject>Parametric models</subject><subject>Photons</subject><subject>Pixels</subject><subject>posterior concentration</subject><subject>Probability distribution</subject><subject>SPECT</subject><subject>Statistical inference</subject><subject>Studies</subject><subject>Tomography</subject><subject>total variation distance</subject><subject>variance estimation in mixed models</subject><issn>0090-5364</issn><issn>2168-8966</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><recordid>eNo9kE1Lw0AQhhdRsFYv3oWANyG6m9mP5GZs17aQJpC0Xpc0uwsN1ehuevDfG2noZV4YHp4ZXoTuCX4mEaEvhIZpUZEIkgs0iQiPwzjh_BJNME5wyIDTa3TjfYsxZgmFCWKbpQzeZJlXG7nKw48iD9arSlbBsC9KuQ7SfB7kRV7KxTZLy2BdzGVW3aIrWx-8uRtzirbvcjNbhlmxWM3SLGwoYX3IdcNMTTW2OyGAw06YxFiGsdUAQAURGrNGM81jypoatLUGTAxMs2RnEwNT9HryfruuNU1vjs1hr9W323_W7ld19V7Nttm4HaPuvCKUYEoxj8SgeDwrfo7G96rtju5r-FoREQs8VDKMKXo6UY3rvHfGnm8QrP6bHZRqbHaAH05w6_vOnUkKjHGgHP4AgNZvfw</recordid><startdate>20141001</startdate><enddate>20141001</enddate><creator>Bochkina, Natalia A.</creator><creator>Green, Peter J.</creator><general>Institute of Mathematical Statistics</general><general>The Institute of Mathematical Statistics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope></search><sort><creationdate>20141001</creationdate><title>THE BERNSTEIN-VON MISES THEOREM AND NONREGULAR MODELS</title><author>Bochkina, Natalia A. ; Green, Peter J.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c415t-6dc5ea4d0fb77363b7e9ef500fd3334717d05cd5d6845ca3dffe3e835d59bf9e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>62F12</topic><topic>62F15</topic><topic>Approximate posterior</topic><topic>Approximation</topic><topic>Asymptotic methods</topic><topic>Bayesian analysis</topic><topic>Bayesian inference</topic><topic>Bernstein–von Mises theorem</topic><topic>boundary</topic><topic>Coordinate systems</topic><topic>Inference</topic><topic>Linear models</topic><topic>Mathematical independent variables</topic><topic>Mathematical models</topic><topic>Matrices</topic><topic>nonregular</topic><topic>Parametric models</topic><topic>Photons</topic><topic>Pixels</topic><topic>posterior concentration</topic><topic>Probability distribution</topic><topic>SPECT</topic><topic>Statistical inference</topic><topic>Studies</topic><topic>Tomography</topic><topic>total variation distance</topic><topic>variance estimation in mixed models</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bochkina, Natalia A.</creatorcontrib><creatorcontrib>Green, Peter J.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>The Annals of statistics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bochkina, Natalia A.</au><au>Green, Peter J.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>THE BERNSTEIN-VON MISES THEOREM AND NONREGULAR MODELS</atitle><jtitle>The Annals of statistics</jtitle><date>2014-10-01</date><risdate>2014</risdate><volume>42</volume><issue>5</issue><spage>1850</spage><epage>1878</epage><pages>1850-1878</pages><issn>0090-5364</issn><eissn>2168-8966</eissn><abstract>We study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the "true" solution occurs on the boundary of the parameter space. 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| ispartof | The Annals of statistics, 2014-10, Vol.42 (5), p.1850-1878 |
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| source | Jstor Complete Legacy; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete; JSTOR Mathematics & Statistics |
| subjects | 62F12 62F15 Approximate posterior Approximation Asymptotic methods Bayesian analysis Bayesian inference Bernstein–von Mises theorem boundary Coordinate systems Inference Linear models Mathematical independent variables Mathematical models Matrices nonregular Parametric models Photons Pixels posterior concentration Probability distribution SPECT Statistical inference Studies Tomography total variation distance variance estimation in mixed models |
| title | THE BERNSTEIN-VON MISES THEOREM AND NONREGULAR MODELS |
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