{"id":357875,"date":"2017-01-25T14:23:03","date_gmt":"2017-01-25T22:23:03","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357875"},"modified":"2018-10-16T20:01:47","modified_gmt":"2018-10-17T03:01:47","slug":"reconstruction-trees-exponential-moment-bounds-linear-estimators","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/reconstruction-trees-exponential-moment-bounds-linear-estimators\/","title":{"rendered":"Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators"},"content":{"rendered":"

Consider a Markov chain (<\/span>\u03be<\/span>v<\/span><\/span>)<\/span>v<\/span>\u2208<\/span>V<\/span><\/span><\/span><\/span>\u2208<\/span>[<\/span>k<\/span>]<\/span>V<\/span><\/span><\/span><\/span><\/span> on the infinite b<\/span><\/span><\/span><\/span>-ary tree T<\/span>=<\/span>(<\/span>V<\/span>,<\/span>E<\/span>)<\/span><\/span><\/span><\/span> with irreducible edge transition matrix M<\/span><\/span><\/span><\/span>, where b<\/span>\u2265<\/span>2<\/span><\/span><\/span><\/span>, k<\/span>\u2265<\/span>2<\/span><\/span><\/span><\/span> and [<\/span>k<\/span>]<\/span>=<\/span>{<\/span>1<\/span>,<\/span>.<\/span>.<\/span>.<\/span>,<\/span>k<\/span>}<\/span><\/span><\/span><\/span>. We denote by L<\/span>n<\/span><\/span><\/span><\/span><\/span> the level-n<\/span><\/span><\/span><\/span> vertices of T<\/span><\/span><\/span><\/span>. Assume M<\/span><\/span><\/span><\/span> has a real second-largest (in absolute value) eigenvalue \u03bb<\/span><\/span><\/span><\/span> with corresponding real eigenvector \u03bd<\/span>\u2260<\/span>0<\/span><\/span><\/span><\/span>. Letting \u03c3<\/span>v<\/span><\/span>=<\/span>\u03bd<\/span>\u03be<\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, we consider the following root-state estimator, which was introduced by Mossel and Peres (2003) in the context of the “recontruction problem” on trees:<\/p>\n

S<\/span>n<\/span><\/span>=<\/span>(<\/span>b<\/span>\u03bb<\/span>)<\/span>\u2212<\/span>n<\/span><\/span><\/span><\/span>\u2211<\/span>x<\/span>\u2208<\/span>L<\/span>n<\/span><\/span><\/span><\/span><\/span>\u03c3<\/span>x<\/span><\/span>.<\/span><\/span><\/span><\/span><\/div>\n

As noted by Mossel and Peres, when b<\/span>\u03bb<\/span>2<\/span><\/span>><\/span>1<\/span><\/span><\/span><\/span> (the so-called Kesten-Stigum reconstruction phase) the quantity S<\/span>n<\/span><\/span><\/span><\/span><\/span> has uniformly bounded variance. Here, we give bounds on the moment-generating functions of S<\/span>n<\/span><\/span><\/span><\/span><\/span> and S<\/span>2<\/span>n<\/span><\/span><\/span><\/span><\/span> when b<\/span>\u03bb<\/span>2<\/span><\/span>><\/span>1<\/span><\/span><\/span><\/span>. Our results have implications for the inference of evolutionary trees.<\/p>\n","protected":false},"excerpt":{"rendered":"

Consider a Markov chain (\u03bev)v\u2208V\u2208[k]V on the infinite b-ary tree T=(V,E) with irreducible edge transition matrix M, where b\u22652, k\u22652 and [k]={1,…,k}. We denote by Ln the level-n vertices of T. Assume M has a real second-largest (in absolute value) eigenvalue \u03bb with corresponding real eigenvector \u03bd\u22600. Letting \u03c3v=\u03bd\u03bev, we consider the following root-state estimator, […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Institute of Mathematical Statistics","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Electronic Communications in Probability","msr_number":"","msr_organization":"","msr_pages_string":"251-261","msr_page_range_start":"251","msr_page_range_end":"261","msr_series":"","msr_volume":"16","msr_copyright":"","msr_conference_name":"","msr_doi":"10.1214\/ECP.v16-1630","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2011-05-19","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/projecteuclid.org\/euclid.ecp\/1465261980","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-357875","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Institute of Mathematical Statistics","msr_edition":"","msr_affiliation":"","msr_published_date":"2011-05-19","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"251-261","msr_chapter":"","msr_isbn":"","msr_journal":"Electronic Communications in Probability","msr_volume":"16","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"357878","msr_publicationurl":"http:\/\/projecteuclid.org\/euclid.ecp\/1465261980","msr_doi":"10.1214\/ECP.v16-1630","msr_publication_uploader":[{"type":"file","title":"0908.2056v1","viewUrl":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-content\/uploads\/2017\/01\/0908.2056v1.pdf","id":357878,"label_id":0},{"type":"url","title":"http:\/\/projecteuclid.org\/euclid.ecp\/1465261980","viewUrl":false,"id":false,"label_id":0},{"type":"doi","title":"10.1214\/ECP.v16-1630","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"http:\/\/projecteuclid.org\/euclid.ecp\/1465261980"}],"msr-author-ordering":[{"type":"user_nicename","value":"peres","user_id":33234,"rest_url":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=peres"},{"type":"text","value":"Sebastien Roch","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"article","related_content":[],"_links":{"self":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/357875","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":1,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/357875\/revisions"}],"predecessor-version":[{"id":519130,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/357875\/revisions\/519130"}],"wp:attachment":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/media?parent=357875"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=357875"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=357875"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=357875"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=357875"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=357875"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=357875"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=357875"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=357875"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=357875"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=357875"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=357875"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=357875"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}