[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84661-en":3,"doc-seo-84661-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84661,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","The Binomial Channel On Capacity Optimal Inputs and Beta-Binomial Approximation","Study of the binomial channel with continuous input alphabet [0,1] and finite output alphabet {0,...,n}. The work characterizes the capacity C(n) and the structure of the capacity-achieving input and induced output distributions, showing optimal inputs are discrete, unique, symmetric around 1/2, and include endpoints 0 and 1. Explicit nonasymptotic bounds yield C(n)=1/2 log(n^2/2πe)+o(1). A beta-binomial reference input Xr~Beta(1/2,1/2) provides the matching lower bound and supports sharp support-size and approximation limits.","arXiv :2607 .02683v 1 [ cs .IT] 2 Jul 2026  \nThe Binomial Channel: On Capacity, Optimal Inputs, and Beta-Binomial Approximation  \nAntonino Favano, Mohammadamin Baniasadi, Ian Zieder, Luca Barletta, and Alex Dytso  \nAbstract  \nWe study the binomial channel with input alphabet [0 , 1] and output alphabet {0,..., n} . We investigate its capacity and the structure of the capacity-achieving input and output distributions. Since the output alphabet is finite whereas the input alphabet is continuous, different input distributions may induce the same output distribution; hence, uniqueness and support properties of optimal inputs do not follow from strict concavity arguments.  \nWe first establish structural properties of the capacity-achieving input distribution. In particular, we show that it is discrete, unique, symmetric around 1/2, and contains the endpoints {0, 1} in its support. We also derive location constraints and bounds on the probability masses of support points, and improve the Witsenhausen-type upper bound on the support size from order n to order n/2 .  \nWe derive explicit nonasymptotic upper and lower bounds on capacity C (n) . These bounds imply C (n) = ~~1~~2 log ~~n~~2~~π~~e + o(1) . The lower bound is obtained by evaluating the mutual information at the reference input Xr ∼ Beta(1/2, 1/2), which induces a beta-binomial output distribution, while the upper bound follows from a minimax redundancy construction.  \nFinally, we prove an improved lower bound on the support size of the capacity-achieving input distribution. We show that the beta-binomial output induced by Xr is asymptotically optimal and close to the capacity-achieving output distribution in relative entropy and χ2 divergence. We also prove a finite-mixture approximation lower bound showing that the beta-binomial output cannot be approximated too accurately by binomial mixtures with few components. Combining these results yields a support-size lower bound of order Ω( √n log log n), with explicit constants. Numerical results illustrate the capacity bounds and optimal input.  \nI. INTRODUCTION  \nWe consider a channel for which the relationship between the input X ∈ [0 , 1] and the output Y ∈ {0,..., n} is described by the binomial distribution:  \nPY |X (y|x) = 􀀒ny􀀓 xy (1 − x)n−y . (1)  \nIn (1), we use the convention that 00 = 1 .  \nThis is an extension of arXiv:2401.12818 . Antonino Favano and Luca Barletta are with the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133 Milano, Italy (e-mail: {antonino.favano, [luca.barletta](luca.barletta}@polimi.it)[}](luca.barletta}@polimi.it)[@polimi.it](luca.barletta}@polimi.it)). Mohammadamin Baniasadi is with the University of California, Davis, CA, USA ([e-mail: mbaniasadi@ucdavis.edu](e-mail: mbaniasadi@ucdavis.edu)). Ian Zieder is with MECA Electronics, Denville, NJ, USA (e-mail: [izieder@e-meca.com](izieder@e-meca.com)). Alex Dytso is with Qualcomm Flarion Technologies, Bridgewater, NJ, USA (e-mail: [odytso@gmail.com](odytso@gmail.com)).  \nJuly 7, 2026 DRAFT  \nThe objective of this paper is twofold. First, we study the capacity C (n) of the binomial channel as a function of the number of trials n, that is  \nC (n) = max I (X;Y ), (2)  \nPX : X∈[0 , 1]  \nand derive explicit nonasymptotic upper and lower bounds whose gap vanishes as n → ∞ . Second, we investigate the structure of the capacity-achieving input distribution (CAID) PX ⋆ . In particular, we study discreteness, uniqueness, symmetry, endpoint optimality, the location and probabilities of the support points, and upper and lower boundson the cardinality of the support. Along the way, we also derive several related results, including properties of beta-binomial output distributions, estimation-theoretic identities for the binomial channel, and a finite-mixture approximation lower bound tailored to binomial mixtures. Together, these results give a more detailed picture of both the capacity and the optimizer for the binomial cha","cbCaikRKteXqD6ud","https://ap.wps.com/l/cbCaikRKteXqD6ud","pdf",728415,1,50,"English","en",105,"# Abstract\n# Introduction\n## Outline and Contributions\n# Channel Model and Objective\n## Notation and KKT Conditions\n## Estimation-Theoretic Identities and Reference Input\n# Finite-Mixture Approximation Results\n## Parseval-Type χ2-Divergence Identity\n## Moment-Matrix Rank Argument\n# Capacity Bounds\n## Reference-Input Lower Bound\n## Minimax Redundancy Upper Bound\n# Structure of Capacity-Achieving Inputs\n## Discreteness, Uniqueness, and Symmetry","[{\"question\":\"What are the input and output alphabets of the binomial channel studied in the paper?\",\"answer\":\"The input X takes values in the continuous alphabet [0,1], while the output Y takes values in the finite set {0,...,n} described by the binomial law P_{Y|X}(y|x)=C(n,y)x^y(1-x)^{n-y}.\"},{\"question\":\"What key structural properties are proved for the capacity-achieving input distribution?\",\"answer\":\"The capacity-achieving input is shown to be discrete and unique, symmetric around 1/2, and to include the endpoints 0 and 1 in its support, together with constraints on the locations and probabilities of support points.\"},{\"question\":\"How does the paper use the beta-binomial approximation to bound capacity and analyze optimal outputs?\",\"answer\":\"Evaluating mutual information under the reference input Xr~Beta(1/2,1/2) yields a nonasymptotic lower bound on capacity and induces a beta-binomial output that is asymptotically optimal. The paper further proves approximation limits showing that few-component binomial mixtures cannot match the beta-binomial output too accurately.\"}]",1784197538,126,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"the-binomial-channel-on-capacity-optimal-inputs-and-beta-binomial-approximation","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/the-binomial-channel-on-capacity-optimal-inputs-and-beta-binomial-approximation/84661/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are the input and output alphabets of the binomial channel studied in the paper?","Question",{"text":75,"@type":76},"The input X takes values in the continuous alphabet [0,1], while the output Y takes values in the finite set {0,...,n} described by the binomial law P_{Y|X}(y|x)=C(n,y)x^y(1-x)^{n-y}.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What key structural properties are proved for the capacity-achieving input distribution?",{"text":80,"@type":76},"The capacity-achieving input is shown to be discrete and unique, symmetric around 1/2, and to include the endpoints 0 and 1 in its support, together with constraints on the locations and probabilities of support points.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper use the beta-binomial approximation to bound capacity and analyze optimal outputs?",{"text":84,"@type":76},"Evaluating mutual information under the reference input Xr~Beta(1/2,1/2) yields a nonasymptotic lower bound on capacity and induces a beta-binomial output that is asymptotically optimal. The paper further proves approximation limits showing that few-component binomial mixtures cannot match the beta-binomial output too accurately.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":21,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]