[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-133721-en":3,"doc-seo-133721-105":31,"detail-sidebar-cat-0-en-105":92},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},133721,1099523885336,"Taylor Morgan","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Statistica - Introduzione alle statistiche bayesiane","Introduzione rapida alle statistiche bayesiane con una ricostruzione storica e un percorso operativo: dalle probabilità dirette alle probabilità inverse. Il materiale collega lo sviluppo del calcolo delle probabilità ai problemi classici, spiegando perché la prospettiva bayesiana differisca da quella frequentista. Sono presentati esempi su probabilità elementare, combinatoria e problemi sulle cause tramite il teorema di Bayes, con cenni a regressione bayesiana e test di ipotesi bayesiani.","Statistica  \nA quick introduction to Bayesian statistics  \nLeonardo Egidi  \nlegidi@units.it  \n[Dipt.to](Dipt.to) di Scienze Economiche, Aziendali, Matematiche e Statistiche, Univ. di Trieste  \nA.A. 2022/23  \nL. Egidi (UNITS) Introduction 1 / 51  \n􀀏 history 􀀏 regression 􀀏 Bayesian hypothesis testing 􀀏  \nIndice  \n1 Historical introduction: from direct to inverse probabilities  \n2 Bayesian regression: foundations and examples  \n3 Bayesian hypothesis testing  \nL. Egidi (UNITS) Introduction 2 / 51  \n􀀏 history 􀀏 regression 􀀏 Bayesian hypothesis testing 􀀏  \nWhy starting from history  \nWe will start with an overview of the history of Bayesian statistics: its development from probability calculus and its relationship with the socalled classical statistics.  \nThis is useful to better understand the di✛erences between Bayesian and classical statistics.  \n\n| 10\u003Cbr>9 8\u003Cbr>7\u003Cbr>6 5\u003Cbr>4\u003Cbr>3\u003Cbr>2\u003Cbr>1 | . Neyman S Pearson . . R. A. Fisher W.S. Gosset (Stu dent) K. Pearson F. Galt on D.V. L indley P−S Laplace L.J. Sav age T. Bayes [B. de](B. de) Fine\u003Cbr>tti A. De M oivre F. P. Ramse y G. Cardano J. Bernoull H . Jeffreys\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>i\u003Cbr>J\u003Cbr>E\u003Cbr>● Liber de ludo aleae (written) ● Bayes T. (Bayes) ●Regression\u003Cbr>● Binomial ● Bayes T. (Laplace) ● NHST\u003Cbr>●LLN ● Least sq. ● Likelihood  ●●Bin−>NormNalrm ●c●ht−i2test ● MCMC \u003Cbr>1500 1600 1700 1800 1900 2000 |\n| --- | --- |\n\nL. Egidi (UNITS) Introduction 3 / 51  \n􀀏 history 􀀏 regression 􀀏 Bayesian hypothesis testing 􀀏  \nGames of chance are the cradle of probability  \nProbability calculus is initially developed to study games of chance: developing strategies to win in games was of interest to nobles, who were willing to pay scholars for them.  \nFor example Galileo in 1620 wrote a note o✛ering the solution of this issue: suppose three dices are thrown and the three numbers obtained added. What is the probability that the total equals 9?  \nThis circumstances help not only because of the money, but also because of the simple structure of the problems involved.  \nL. Egidi (UNITS) Introduction 4 / 51  \n􀀏 history 􀀏 regression 􀀏 Bayesian hypothesis testing 􀀏  \nElementary probability  \nThe ✜rst examples of probability problems are concerned with simple random mechanisms whose symmetry o✛ered the solution.  \nQ: A marble is randomly drawn from an urn containing R red marbles and W white marbles, what is the probability that the marble is red?  \nA: By symmetry  \nP (red) = R~~ ~~~~ ~~W  \nGames of chances are easily tackled using the ✜rst de✜nition of probability, based on symmetry  \n\\# favourable outcomes  \nprob. of event =  \n\\# possible outcomes  \nL. Egidi (UNITS) Introduction 5 / 51  \n􀀏 history 􀀏 regression 􀀏 Bayesian hypothesis testing 􀀏  \nElementary probability and combinatorics  \nFor more ✏complicated✑ questions tools were developed to count favourable and unfavourable outcomes.  \nQ: We draw a marble from an urn containing R red marbles and W white marbles m times (putting it back in the urn after each draw), what is the probability that r out of m are red?  \nA: Still by symmetry  \nP (r red out of m) = 􀀒 mr􀀓 􀀒 ~~ ~~R~~ ~~~~ ~~W 􀀓 r 􀀒 1 􀀀 R~~ ~~~~ ~~W 􀀓 m􀀀r  \nGirolamo Cardano (1501-1576), wrote the ✜rst systematic treatment of probability in 1576: Liber de ludo aleae; this, however was not published until 1663 . He was a polymath with interests ranging from mathematics to biology. He was also a gambler (anda rumor exists that he did not publish his book on probability because his knowledge gave him an advantage in betting) .  \nL. Egidi (UNITS) Introduction 6 / 51  \n􀀏 history 􀀏 regression 􀀏 Bayesian hypothesis testing 􀀏  \nIndirect problems: the probability of causes  \nWithin the above experiment, we can also ask the following question Having observed X = x, what is the probability that the urn contains R red marbles?  \nThis is solved by Bayes theorem.  \nThomas Bayes (c. 1702-1761) was a Presbyterian minister. In Essay Towards Solving a Problem in the Doctrine of Chances (1763) he considers the inverse pro","cbCaiadleeEJDT8B","https://ap.wps.com/l/cbCaiadleeEJDT8B","pdf",983516,3,1,56,"English","en",105,"# Historical introduction: from direct to inverse probabilities\n# Bayesian regression: foundations and examples\n# Bayesian hypothesis testing","[{\"question\":\"Qual è l’obiettivo dell’introduzione alle statistiche bayesiane?\",\"answer\":\"Fornire una visione storica e concettuale che chiarisca le differenze tra probabilità dirette e inverse e prepari all’uso di strumenti bayesiani per modelli e inferenza.\"},{\"question\":\"Come si risolve un problema di probabilità della causa dopo aver osservato un risultato?\",\"answer\":\"Si applica il teorema di Bayes per convertire l’informazione osservata in probabilità sulle ipotesi/cause, come nell’esempio dell’urna con marbles rosse.\"},{\"question\":\"Quali argomenti vengono trattati oltre al teorema di Bayes?\",\"answer\":\"Il percorso include fondamenti ed esempi di regressione bayesiana e la logica dei test di ipotesi bayesiani, oltre a richiami a probabilità elementare e combinatoria.\"}]","Statistica - Introduzione alle statistiche bayesiane | PDF",1787225701,141,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"statistics-introduction-to-bayesian-statistics","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,51],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":20},"https://docshare.wps.com/document/research-report/",{"item":52,"name":13,"@type":44,"position":53},"https://docshare.wps.com/document/statistics-introduction-to-bayesian-statistics/133721/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-27","2026-08-20",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Qual è l’obiettivo dell’introduzione alle statistiche bayesiane?","Question",{"text":76,"@type":77},"Fornire una visione storica e concettuale che chiarisca le differenze tra probabilità dirette e inverse e prepari all’uso di strumenti bayesiani per modelli e inferenza.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Come si risolve un problema di probabilità della causa dopo aver osservato un risultato?",{"text":81,"@type":77},"Si applica il teorema di Bayes per convertire l’informazione osservata in probabilità sulle ipotesi/cause, come nell’esempio dell’urna con marbles rosse.",{"name":83,"@type":74,"acceptedAnswer":84},"Quali argomenti vengono trattati oltre al teorema di Bayes?",{"text":85,"@type":77},"Il percorso include fondamenti ed esempi di regressione bayesiana e la logica dei test di ipotesi bayesiani, oltre a richiami a probabilità elementare e combinatoria.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]