NONPARAMETRIC BAYESIAN ANALYSIS OF THE COMPOUND POISSON PRIOR FOR SUPPORT BOUNDARY RECOVERY
Given data from a Poisson point process with intensity (x, y) ↦ n₁ (f(x) ≤ y), frequentist properties for the Bayesian reconstruction of the support boundary function f are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly...
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| Veröffentlicht in: | The Annals of statistics 2020-06, Vol.48 (3), p.1432-1451 |
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| container_title | The Annals of statistics |
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| creator | Reiss, Markus Schmidt-Hieber, Johannes |
| description | Given data from a Poisson point process with intensity (x, y) ↦ n₁ (f(x) ≤ y), frequentist properties for the Bayesian reconstruction of the support boundary function f are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone support boundaries and adapts to Hölder smooth boundaries. We then derive a limiting shape result for a compound Poisson process prior and a function space with increasing parameter dimension. It is shown that the marginal posterior of the mean functional performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, (1 – α)-credible sets are also asymptotic (1 – α)-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions f outside the function class. |
| doi_str_mv | 10.1214/19-AOS1853 |
| format | Article |
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We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone support boundaries and adapts to Hölder smooth boundaries. We then derive a limiting shape result for a compound Poisson process prior and a function space with increasing parameter dimension. It is shown that the marginal posterior of the mean functional performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, (1 – α)-credible sets are also asymptotic (1 – α)-confidence intervals. 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We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone support boundaries and adapts to Hölder smooth boundaries. We then derive a limiting shape result for a compound Poisson process prior and a function space with increasing parameter dimension. It is shown that the marginal posterior of the mean functional performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, (1 – α)-credible sets are also asymptotic (1 – α)-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions f outside the function class.</description><subject>Bayesian analysis</subject><subject>Boundary conditions</subject><subject>Confidence intervals</subject><subject>Contracts</subject><subject>Data recovery</subject><subject>Function space</subject><subject>Linear functions</subject><subject>Nonparametric statistics</subject><subject>Poisson density functions</subject><subject>Poisson distribution</subject><subject>Smooth boundaries</subject><subject>Statistical analysis</subject><issn>0090-5364</issn><issn>2168-8966</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNo90MFLwzAUBvAgCs7pxbsQ8CZU85ImTY5Z7Vxha0raCcVDSdcWHGq13Q7-93ZseHi8w_vxPfgQugXyCBT8J1CeNhlIzs7QhIKQnlRCnKMJIYp4nAn_El0Nw5YQwpXPJugtMUmqrV5FuY1DPNNFlMU6wTrRyyKLM2zmOF9EODSr1KyTZ5yaOMtMglMbG4vn42TrNDU2x7PDXdsC2yg0r5EtrtFF6z6G5ua0p2g9j_Jw4S3NSxzqpbdhEOw8J0XTMgcb5_yG-1RQJRrwA9LWQJjiVe0kDwInXFsFPuGSC0KAQF1VTd0qyabo_pj73Xc_-2bYldtu33-NL0vqMyBUckJH9XBUm74bhr5py-_-_dP1vyWQ8lBeCao8lTfiuyPeDruu_5dUKAYcAvYHjf5ikQ</recordid><startdate>20200601</startdate><enddate>20200601</enddate><creator>Reiss, Markus</creator><creator>Schmidt-Hieber, Johannes</creator><general>Institute of Mathematical Statistics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope></search><sort><creationdate>20200601</creationdate><title>NONPARAMETRIC BAYESIAN ANALYSIS OF THE COMPOUND POISSON PRIOR FOR SUPPORT BOUNDARY RECOVERY</title><author>Reiss, Markus ; Schmidt-Hieber, Johannes</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c317t-a86ef3a1caa4e5426296e1470fd10395bda8577a6afb740585600101dbbedf983</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Bayesian analysis</topic><topic>Boundary conditions</topic><topic>Confidence intervals</topic><topic>Contracts</topic><topic>Data recovery</topic><topic>Function space</topic><topic>Linear functions</topic><topic>Nonparametric statistics</topic><topic>Poisson density functions</topic><topic>Poisson distribution</topic><topic>Smooth boundaries</topic><topic>Statistical analysis</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Reiss, Markus</creatorcontrib><creatorcontrib>Schmidt-Hieber, Johannes</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>Reiss, Markus</au><au>Schmidt-Hieber, Johannes</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>NONPARAMETRIC BAYESIAN ANALYSIS OF THE COMPOUND POISSON PRIOR FOR SUPPORT BOUNDARY RECOVERY</atitle><jtitle>The Annals of statistics</jtitle><date>2020-06-01</date><risdate>2020</risdate><volume>48</volume><issue>3</issue><spage>1432</spage><epage>1451</epage><pages>1432-1451</pages><issn>0090-5364</issn><eissn>2168-8966</eissn><abstract>Given data from a Poisson point process with intensity (x, y) ↦ n₁ (f(x) ≤ y), frequentist properties for the Bayesian reconstruction of the support boundary function f are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone support boundaries and adapts to Hölder smooth boundaries. We then derive a limiting shape result for a compound Poisson process prior and a function space with increasing parameter dimension. It is shown that the marginal posterior of the mean functional performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, (1 – α)-credible sets are also asymptotic (1 – α)-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions f outside the function class.</abstract><cop>Hayward</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/19-AOS1853</doi><tpages>20</tpages><oa>free_for_read</oa></addata></record> |
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| source | JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete |
| subjects | Bayesian analysis Boundary conditions Confidence intervals Contracts Data recovery Function space Linear functions Nonparametric statistics Poisson density functions Poisson distribution Smooth boundaries Statistical analysis |
| title | NONPARAMETRIC BAYESIAN ANALYSIS OF THE COMPOUND POISSON PRIOR FOR SUPPORT BOUNDARY RECOVERY |
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