{"id":39,"date":"2024-04-04T00:12:02","date_gmt":"2024-04-03T21:12:02","guid":{"rendered":"https:\/\/sisu.ut.ee\/samm\/bayesi-statistika\/"},"modified":"2025-09-18T16:27:47","modified_gmt":"2025-09-18T13:27:47","slug":"bayesi-statistika","status":"publish","type":"page","link":"https:\/\/sisu.ut.ee\/samm\/bayesi-statistika\/","title":{"rendered":"Bayesi statistika"},"content":{"rendered":"<p style=\"text-align: right;\"><strong>Taavi Unt<br>2020<\/strong><\/p>\n<p>Statistikas on eristatavad kaks peamist koolkonda: Bayesi statistika ja sagedusstatistika ehk nn klassikaline statistika.\u00a0<span lang=\"ET\"><span style=\"line-height: 107%;\">M\u00f5lemal juhul on eesm\u00e4rk olemasolevate vaatluste (<a href=\"https:\/\/samm.ut.ee\/valimi-moodustamine\/\">valimi<\/a>) pealt hinnata kogu populatsioonis kehtivaid seoseid v\u00f5i huvipakkuvaid n\u00e4itajaid. Teisis\u00f5nu soovitakse hinnata parameetrit <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">(<i>theta<\/i>),<\/span><\/span><span lang=\"ET\"><span style=\"line-height: 107%;\"> mis v\u00f5ib olla nii arv (nt \u00fcldkogumi keskmine) kui vektor (nt regressioonanal\u00fc\u00fcsi kordajad). Milles seisneb aga Bayesi ja sagedusstatistika erinevus? Erinevused tulenevad eelk\u00f5ige filosoofilisest l\u00e4henemisest.<\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Sagedusstatistikud l\u00e4htuvad eeldusest, et andmete saamise protsess on korratav. Lisaks eeldatakse, et huvipakkuv parameeter <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">ei ole k\u00fcll teada, kuid on fikseeritud (konstant), ning vaatlusandmed on ainus teave, mida parameetri <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">hindamisel kasutada.<\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Bayesi statistika seevastu ei n\u00f5ua, et andmete genereerimise protsess oleks korratav. Kuigi l\u00e4hte-eelduste kohaselt ei ole parameetri <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">v\u00e4\u00e4rtus teada, eeldatakse see olevat kirjeldatav t\u00f5en\u00e4osuslikult (jaotusega). Parameetri <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">kirjeldamisel kasutatakse nii vaatlusandmeid kui ka anal\u00fc\u00fcsi l\u00e4biviija eelteadmisi parameetri kohta. Seejuures andmete abil t\u00e4psustatakse eelteadmisi.<\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Nii Bayesi kui sagedusstatistika puhul eeldatakse, et vaadeldavad andmed <\/span><\/span> <b><i><span lang=\"ET\"><span style=\"line-height: 107%;\">x\u00a0<\/span><\/span><\/i><\/b><span lang=\"ET\"><span style=\"line-height: 107%;\">on mingist parameetriga <\/span><\/span> <i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">m\u00e4\u00e4ratud jaotusest. Seda, kui h\u00e4sti antud andmed erinevate parameetri <\/span><\/span> <i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">v\u00e4\u00e4rtustega koosk\u00f5las on, kirjeldab t\u00f5ep\u00e4rafunktsioon <\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"62\" height=\"38\" class=\"alignnone wp-image-371\" title=\"screen_shot_2020-04-06_at_15.08.40.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.08.40.png\" alt=\"valem\"><span lang=\"ET\"><span style=\"line-height: 107%;\">. Fikseeritud <\/span><\/span> <i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8<\/span><\/span><\/i> <span lang=\"EN-US\"><span style=\"line-height: 107%;\"><span style=\"position: relative;\"><span style=\"top: 4.0pt;\"><img decoding=\"async\" id=\"_x0000_i1025\" style=\"width: 7pt; height: 15pt;\" src=\"\/\/\/\/Users\/sandrasaar\/Library\/Group%20Containers\/UBF8T346G9.Office\/TemporaryItems\/msohtmlclip\/clip_image010.emz\"> <\/span><\/span><\/span><\/span><span lang=\"ET\"><span style=\"line-height: 107%;\">\u00a0korral nimetatakse funktsiooni <\/span><\/span> <i><span lang=\"ET\"><span style=\"line-height: 107%;\">f<\/span><\/span><\/i>\u00a0\u00a0<span lang=\"ET\"><span style=\"line-height: 107%;\">tihedusfunktsiooniks, kui vaatlused on pidevast jaotusest, v\u00f5i t\u00f5en\u00e4osusfunktsiooniks, kui vaatused on diskreetsest jaotusest (vt <\/span><\/span><a title=\"\" href=\"http:\/\/samm.ut.ee\/tunnused-ja-nende-tyybid\" target=\"_blank\" rel=\"noopener\" data-url=\"http:\/\/samm.ut.ee\/tunnused-ja-nende-tyybid\">peat\u00fckki tunnustest ja nende t\u00fc\u00fcpidest<\/a><span lang=\"ET\"><span style=\"line-height: 107%;\">).\u00a0<\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Kui sagedusstatistika kohaselt on <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">mingi tundmatu arv, siis Bayesi statistikas eeldatakse, et\u00a0<\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">on juhuslik suurus. Eelteadmist parameetri <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">kohta v\u00e4ljendatakse parameetri eeljaotusega <\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"48\" height=\"34\" class=\"alignnone wp-image-372\" title=\"screen_shot_2020-04-06_at_15.09.14.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.09.14.png\" alt=\"p\"><i><span lang=\"ET\"><span style=\"line-height: 107%;\">.<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\"> Eeljaotus v\u00f5ib seejuures s\u00f5ltuda veel omakorda parameetri(te)st <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b1<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">. Tavaliselt eeljaotust t\u00e4histades seda ei r\u00f5hutata, kuid kui on vaja sellele siiski t\u00e4helepanu juhtida, v\u00f5ib eeljaotust t\u00e4histada kujul <\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"61\" height=\"29\" class=\"alignnone wp-image-373\" title=\"screen_shot_2020-04-06_at_15.09.56.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.09.56.png\" alt=\"p\"><span lang=\"ET\"><span style=\"line-height: 107%;\">. Parameetrit <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b1\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">nimetatakse h\u00fcperparameetriks ning see fikseeritakse anal\u00fc\u00fcsi l\u00e4biviija poolt vastavalt tema eelteadmistele uuritava parameetri <\/span><\/span><i><span lang=\"ET\"><span style=\"line-height: 107%;\">\u03b8\u00a0<\/span><\/span><\/i><span lang=\"ET\"><span style=\"line-height: 107%;\">kohta.<\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Selleks, et kombineerida anal\u00fc\u00fctiku eelteadmisi (<\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"61\" height=\"29\" class=\"alignnone wp-image-374\" title=\"screen_shot_2020-04-06_at_15.09.56_01.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.09.56_01.png\" alt=\"p\"><span lang=\"EN-US\"><span style=\"line-height: 107%;\"><span style=\"position: relative;\"><span style=\"top: 4.0pt;\">\u00a0<\/span><\/span><\/span><\/span><span lang=\"ET\"><span style=\"line-height: 107%;\">) ja andmetes sisalduvat informatsiooni (<\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"52\" height=\"39\" class=\"alignnone wp-image-375\" title=\"screen_shot_2020-04-06_at_15.10.59.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.10.59.png\" alt=\"f\"><span lang=\"EN-US\"><span style=\"line-height: 107%;\"><span style=\"position: relative;\"><span style=\"top: 4.0pt;\">\u00a0<\/span><\/span><\/span><\/span><span lang=\"ET\"><span style=\"line-height: 107%;\">), tuleb kasutada Bayesi valemit:<\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"234\" height=\"80\" class=\"wp-image-290 aligncenter\" title=\"screen_shot_2020-03-30_at_11.17.16.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-03-30_at_11.17.16.png\" alt=\"bayesi valem\"><\/p>\n<p>kus\u00a0 funktsiooni <img loading=\"lazy\" decoding=\"async\" width=\"61\" height=\"29\" class=\"alignnone wp-image-376\" title=\"screen_shot_2020-04-06_at_15.11.36.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.11.36.png\" alt=\"valem\"><!-- [if gte msEquation 12]&gt;--><i><span lang=\"ET\" style=\"font-size: 12.0pt; line-height: 107%; font-family: 'Cambria Math',serif;\">p<\/span><\/i><i><span lang=\"ET\" style=\"font-size: 12.0pt; line-height: 107%; font-family: 'Cambria Math',serif;\">\u03b8<\/span><\/i><b><i><span lang=\"ET\" style=\"font-size: 12.0pt; line-height: 107%; font-family: 'Cambria Math',serif;\">x<\/span><\/i><\/b> \u00a0nimetatakse parameetri\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"18\" height=\"27\" class=\"alignnone wp-image-377\" title=\"screen_shot_2020-04-06_at_15.12.10.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.12.10.png\" alt=\"screen_shot_2020-04-06_at_15.12.10.png\">\u00a0j\u00e4reljaotuseks. Funktsioon\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"45\" height=\"35\" class=\"alignnone wp-image-378\" title=\"screen_shot_2020-04-06_at_15.12.50.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.12.50.png\" alt=\"valem\"> on andmete tihedusfunktsioon \u00fcle k\u00f5ikv\u00f5imalike <img loading=\"lazy\" decoding=\"async\" width=\"18\" height=\"27\" class=\"alignnone wp-image-377\" title=\"screen_shot_2020-04-06_at_15.12.10.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.12.10.png\" alt=\"screen_shot_2020-04-06_at_15.12.10.png\"><!-- [if gte msEquation 12]&gt;--><i><span lang=\"ET\" style=\"font-size: 12.0pt; line-height: 107%; font-family: 'Cambria Math',serif;\">\u03b8<\/span><\/i> \u00a0v\u00e4\u00e4rtuste, st\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"216\" height=\"33\" class=\"alignnone wp-image-291\" title=\"screen_shot_2020-03-30_at_11.17.59.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-03-30_at_11.17.59.png\" alt=\"valem\"><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Seega on meil olemas eeskiri, kuidas olemasolevaid andmeid arvesse v\u00f5ttes oma eelteadmisi t\u00e4psustada. Seejuures on sama eeskirja v\u00f5imalik t\u00e4iendavate andmete laekumisel korduvalt rakendada \u2013 selleks tuleb esmaste andmete korral saadud j\u00e4reljaotust kasutada uute andmete saamisel eeljaotusena. Intuitiivselt v\u00f5ib sellele m\u00f5elda selliselt, et enne uuringu l\u00e4biviimist oli meil mingi arvamus uuritava parameetri jaotuse kohta. P\u00e4rast esimese komplekti andmete n\u00e4gemist meie arvamus t\u00e4psustus. Teise komplekti andmete k\u00e4ttesaamisel saame seda kasutada kui parimat teavet parameetri jaotuse kohta. Sama m\u00f5ttek\u00e4iku saab korrata iga kord, kui saabub uus komplekt andmeid. \u00dchtlasi v\u00f5ib sellisele protseduurile m\u00f5elda kui \u00f5ppimisprotsessile, kus uute faktide ilmnemisel uuendame oma teadmisi.<\/span><\/span><\/p>\n<p><\/p><div class=\"accordion mb-3\">\n        <div class=\"accordion-item accordion-item--white\">\n        <h2 class=\"accordion-header\" id=\"accordion-69de8869e1d1d-heading\">\n            <button class=\"accordion-button collapsed\" type=\"button\" data-bs-toggle=\"collapse\" data-bs-target=\"#accordion-69de8869e1d1d-collapse\" aria-expanded=\"true\" aria-controls=\"accordion-69de8869e1d1d-collapse\">Punkthinnang<\/button>\n        <\/h2>\n        <div id=\"accordion-69de8869e1d1d-collapse\" class=\"accordion-collapse collapse\" aria-labelledby=\"accordion-69de8869e1d1d-heading\">\n            <div class=\"accordion-body\">\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Bayesi l\u00e4henemist kasutades on anal\u00fc\u00fcsi tulemuseks uuritava parameetri jaotuse hinnang. Olenevalt olukorrast v\u00f5ib jaotuse raporteerimine olla aga liialt inforohke ja m\u00f5nele sihtr\u00fchmale ka raskesti m\u00f5istetav. Tihtipeale soovitakse tulemuseks \u201e\u00fchte numbrit\u201c: anal\u00fc\u00fcsi kontekstis punkthinnangut. Selle k\u00e4igus l\u00e4heb k\u00fcll kaduma palju teavet, kuid muudab tulemuste esitamise lihtsamaks. <\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">T\u00fc\u00fcpilisteks punkthinnanguteks on kas j\u00e4reljaotuse keskv\u00e4\u00e4rtus, mediaan v\u00f5i mood. Teades j\u00e4reljaotust, on v\u00f5imalik v\u00e4lja arvutada ka selle muid karakteristikuid, nagu n\u00e4iteks standardh\u00e4lvet v\u00f5i kvantiile. <\/span><\/span><\/p>\n<p><\/p><\/div>\n        <\/div>\n        <\/div>\n    <\/div>\n<p><\/p><div class=\"accordion mb-3\">\n        <div class=\"accordion-item accordion-item--white\">\n        <h2 class=\"accordion-header\" id=\"accordion-69de8869e1d5f-heading\">\n            <button class=\"accordion-button collapsed\" type=\"button\" data-bs-toggle=\"collapse\" data-bs-target=\"#accordion-69de8869e1d5f-collapse\" aria-expanded=\"true\" aria-controls=\"accordion-69de8869e1d5f-collapse\">N\u00e4ide \u2013 osakaalu hindamine(valitsuse toetusm\u00e4\u00e4r)<\/button>\n        <\/h2>\n        <div id=\"accordion-69de8869e1d5f-collapse\" class=\"accordion-collapse collapse\" aria-labelledby=\"accordion-69de8869e1d5f-heading\">\n            <div class=\"accordion-body\">\n<p>Olgu eesm\u00e4rgiks hinnata, kui suur osa elanikkonnast toetab praegust valitsust. Selleks k\u00fcsitakse <!-- [if gte msEquation 12]&gt;--><i><span lang=\"ET\" style=\"font-size: 12.0pt; line-height: 107%; font-family: 'Cambria Math',serif;\">N<\/span><\/i> \u00a0isikult, kas ta toetab praegust valitust v\u00f5i mitte. Saadud andmed on seega <img loading=\"lazy\" decoding=\"async\" width=\"169\" height=\"32\" class=\"alignnone wp-image-292\" title=\"screen_shot_2020-03-30_at_11.19.30.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-03-30_at_11.19.30.png\" alt=\"valem\">\u00a0kus\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"56\" height=\"21\" class=\"alignnone wp-image-293\" title=\"screen_shot_2020-03-30_at_11.20.02.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-03-30_at_11.20.02.png\" alt=\"valem\"><span lang=\"ET\"><span style=\"line-height: 107%;\">, kui i-s vastanu ei toeta valitsust, ja\u00a0<\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"59\" height=\"30\" class=\"alignnone wp-image-294\" title=\"screen_shot_2020-03-30_at_11.20.58.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-03-30_at_11.20.58.png\" alt=\"valem\">,\u00a0kui toetab. T\u00e4histagu\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"22\" height=\"27\" class=\"alignnone wp-image-295\" title=\"screen_shot_2020-03-30_at_11.22.14.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-03-30_at_11.22.14.png\" alt=\"m\u00e4rk\">\u00a0uuritavat toetusm\u00e4\u00e4ra.<\/p>\n<p>Sagedusstatistikud leiaksid hinnangu toetusm\u00e4\u00e4rale selliselt, et loevad kokku, mitu vastanut on toetaval seisukohal ja jagavad saadud arvu valimimahuga. Kuna <img loading=\"lazy\" decoding=\"async\" width=\"111\" height=\"32\" class=\"alignnone wp-image-296\" title=\"screen_shot_2020-03-30_at_11.22.52.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-03-30_at_11.22.52.png\" alt=\"valem\">t\u00e4histab toetajate arvu, siis <img loading=\"lazy\" decoding=\"async\" width=\"67\" height=\"40\" class=\"alignnone wp-image-297\" title=\"screen_shot_2020-03-30_at_11.23.21.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-03-30_at_11.23.21.png\" alt=\"valem\"><\/p>\n<p>Bayesi statistikud lahendaksid \u00fclesande aga j\u00e4rgmiselt. Esmalt paneme t\u00e4hele, et kui valitsuse toetusm\u00e4\u00e4r elanikkonna seas on<!-- [if gte msEquation 12]&gt;--><i><span lang=\"ET\" style=\"font-size: 12.0pt; line-height: 107%; font-family: 'Cambria Math',serif;\"> \u03b8<\/span><\/i> , siis valides \u00fche juhusliku isiku, on ta t\u00f5en\u00e4osusega <img loading=\"lazy\" decoding=\"async\" width=\"23\" height=\"25\" class=\"alignnone wp-image-298\" title=\"screen_shot_2020-04-06_at_09.07.47.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.07.47.png\" alt=\"o\">valitsust toetav. Seega, valides <em>N\u00a0<\/em>isikut, on toetajate arv\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"27\" height=\"31\" class=\"alignnone wp-image-299\" title=\"screen_shot_2020-04-06_at_09.08.52.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.08.52.png\" alt=\"Nt\">\u00a0binoomjaotusega: <img loading=\"lazy\" decoding=\"async\" width=\"124\" height=\"35\" class=\"alignnone wp-image-300\" title=\"screen_shot_2020-04-06_at_09.09.22.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.09.22.png\" alt=\"valem\">J\u00e4relikult andmete t\u00f5ep\u00e4rafunktsioon avaldub j\u00e4rgmiselt:\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"273\" height=\"43\" class=\"wp-image-301 aligncenter\" title=\"screen_shot_2020-04-06_at_09.09.55.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.09.55.png\" alt=\"valem\">kus\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"41\" height=\"45\" class=\"alignnone wp-image-302\" title=\"screen_shot_2020-04-06_at_09.10.23.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.10.23.png\" alt=\"c\">t\u00e4histab kombinatsioonide arvu valimaks <em>N\u00a0<\/em>isiku seast\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"27\" height=\"35\" class=\"alignnone wp-image-303\" title=\"screen_shot_2020-04-06_at_09.12.09.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.12.09.png\" alt=\"Nt\">isikut.<\/p>\n<p>Selleks, et saada parameetri <img loading=\"lazy\" decoding=\"async\" width=\"23\" height=\"25\" class=\"alignnone wp-image-298\" title=\"screen_shot_2020-04-06_at_09.07.47.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.07.47.png\" alt=\"o\"><span style=\"line-height: normal;\"><span lang=\"ET\">j\u00e4reljaotust, on veel vaja esmalt \u00e4ra fikseerida sobiv eeljaotus. Sobiv eeljaotus on nimelt selline, mis on koosk\u00f5las anal\u00fc\u00fcsi l\u00e4biviija teadmistega ega l\u00e4he vastuollu t\u00f5en\u00e4osusteooriaga.\u00a0<\/span><\/span><\/p>\n<p>Valime eeljaotuseks beeta jaotuse. Beeta jaotus on kirjeldatav parameetritega <img loading=\"lazy\" decoding=\"async\" width=\"18\" height=\"24\" class=\"alignnone wp-image-304\" title=\"screen_shot_2020-04-06_at_09.13.21.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.21.png\" alt=\"a\">ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"19\" height=\"28\" class=\"alignnone wp-image-305\" title=\"screen_shot_2020-04-06_at_09.13.52.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.52.png\" alt=\"b\"><span style=\"line-height: normal;\"><span lang=\"ET\">ning selle tihedusfunktsioon avaldub kujul <\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"398\" height=\"76\" class=\"wp-image-306 aligncenter\" title=\"screen_shot_2020-04-06_at_09.14.29.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.14.29.png\" alt=\"valem\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.14.29.png 398w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.14.29-300x57.png 300w\" sizes=\"auto, (max-width: 398px) 100vw, 398px\">kus\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"18\" height=\"26\" class=\"alignnone wp-image-307\" title=\"screen_shot_2020-04-06_at_09.14.59.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.14.59.png\" alt=\"g\">t\u00e4histab gamma funktsiooni. Seejuures\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"18\" height=\"24\" class=\"alignnone wp-image-304\" title=\"screen_shot_2020-04-06_at_09.13.21.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.21.png\" alt=\"a\">ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"19\" height=\"28\" class=\"alignnone wp-image-305\" title=\"screen_shot_2020-04-06_at_09.13.52.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.52.png\" alt=\"b\"><span style=\"line-height: normal;\"><span lang=\"ET\">on h\u00fcperparameetrid, st need m\u00e4\u00e4ratakse anal\u00fc\u00fcsi l\u00e4biviija poolt.<\/span><\/span><\/p>\n<p style=\"margin-bottom: .0001pt;\"><span style=\"line-height: normal;\"><span lang=\"ET\">Kuna beeta jaotusest juhuslik v\u00e4\u00e4rtus saab omada v\u00e4\u00e4rtusi vahemikus 0 kuni 1, siis sobib see t\u00f5en\u00e4osusteoreetiliselt eeljaotuseks valitsuse toetusm\u00e4\u00e4rale, mis samuti j\u00e4\u00e4b vahemikku 0 kuni 1. <\/span><\/span><\/p>\n<p style=\"margin-bottom: .0001pt;\"><span style=\"line-height: normal;\"><span lang=\"ET\">Kasutades Bayesi valemit, leiame toetusm\u00e4\u00e4ra j\u00e4reljaotuse:<\/span><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"471\" height=\"214\" class=\"wp-image-308 aligncenter\" title=\"screen_shot_2020-04-06_at_09.17.19.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.17.19.png\" alt=\"valemid\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.17.19.png 471w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.17.19-300x136.png 300w\" sizes=\"auto, (max-width: 471px) 100vw, 471px\"><\/p>\n<p>Saadud avaldises on parameetrist <img loading=\"lazy\" decoding=\"async\" width=\"23\" height=\"25\" class=\"alignnone wp-image-298\" title=\"screen_shot_2020-04-06_at_09.07.47.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.07.47.png\" alt=\"o\">s\u00f5ltuv osa\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"231\" height=\"39\" class=\"alignnone wp-image-309\" title=\"screen_shot_2020-04-06_at_09.19.08.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.19.08.png\" alt=\"valem\"><span lang=\"ET\"><span style=\"line-height: 107%;\">On n\u00e4ha,\u00a0 et see vastab omakorda beeta jaotusele parameetritega <\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"302\" height=\"47\" class=\"wp-image-310 aligncenter\" title=\"screen_shot_2020-04-06_at_09.25.59.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.25.59.png\" alt=\"valem\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.25.59.png 302w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.25.59-300x47.png 300w\" sizes=\"auto, (max-width: 302px) 100vw, 302px\"><span lang=\"ET\"><span style=\"line-height: 107%;\">Lisaks, saadud avaldises olev komponent\u00a0<\/span><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"182\" height=\"91\" class=\"wp-image-311 aligncenter\" title=\"screen_shot_2020-04-06_at_09.26.29.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.26.29.png\" alt=\"valem\">ei s\u00f5ltu uuritavast parameetrist <img loading=\"lazy\" decoding=\"async\" width=\"23\" height=\"25\" class=\"alignnone wp-image-298\" title=\"screen_shot_2020-04-06_at_09.07.47.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.07.47.png\" alt=\"o\"><span lang=\"ET\"><span style=\"line-height: 107%;\">vaid fikseeritud andmete ja h\u00fcperparameetrite korral on tegemist mingi arvuga. Kuna j\u00e4reljaotuse n\u00e4ol on tegemist tihedusfunktsiooniga, siis ainuv\u00f5imalik j\u00e4reldus on, et<\/span><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"187\" height=\"110\" class=\"wp-image-312 aligncenter\" title=\"screen_shot_2020-04-06_at_09.27.16.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.27.16.png\" alt=\"valem\"><span lang=\"ET\"><span style=\"line-height: 107%;\">on beeta jaotuse normeerivaks konstandiks, st<\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">\u00a0<\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"322\" height=\"94\" class=\"wp-image-313 aligncenter\" title=\"screen_shot_2020-04-06_at_09.27.59.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.27.59.png\" alt=\"valem\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.27.59.png 322w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.27.59-300x88.png 300w\" sizes=\"auto, (max-width: 322px) 100vw, 322px\">ja j\u00e4reljaotuseks on beeta jaotus parameetritega <img loading=\"lazy\" decoding=\"async\" width=\"31\" height=\"28\" class=\"alignnone wp-image-314\" title=\"screen_shot_2020-04-06_at_09.28.38.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.28.38.png\" alt=\"a\">ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"40\" height=\"30\" class=\"alignnone wp-image-315\" title=\"screen_shot_2020-04-06_at_09.29.02.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.29.02.png\" alt=\"b\"><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Asjaolu, et j\u00e4reljaotus on samast jaotusklassist nagu eeljaotus, annab suure arvutusliku eelise. Antud n\u00e4ite korral tuleb j\u00e4reljaotuse leidmiseks vaid vastavalt valemile uuendada eeljaotuse parameetreid. Seejuures saame t\u00e4pselt sama l\u00e4henemist kasutada t\u00e4iendavate andmete laekumisel. Sellist jaotust, mille korral nii eel- kui ka j\u00e4reljaotus on samast klassist, nimetatakse vaatlusandmete jaotuse kaasjaotuseks (<i>conjugate prior<\/i>). Antud n\u00e4itest on n\u00e4ha, et binoomjaotusega vaatluste kaasjaotuseks on beeta jaotus. <\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Kokkuv\u00f5tvalt, eeldades eeljaotusena beeta jaotust, saame valitsuse toetusm\u00e4\u00e4ra j\u00e4reljaotuse arvutamiseks j\u00e4rgmise eeskirja:<\/span><\/span><\/p>\n<p>1. Fikseeri eeljaotuse\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"18\" height=\"24\" class=\"alignnone wp-image-304\" title=\"screen_shot_2020-04-06_at_09.13.21.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.21.png\" alt=\"a\">ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"19\" height=\"28\" class=\"alignnone wp-image-305\" title=\"screen_shot_2020-04-06_at_09.13.52.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.52.png\" alt=\"b\">,<br>2. Arvuta\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"118\" height=\"27\" class=\"alignnone wp-image-316\" title=\"screen_shot_2020-04-06_at_09.52.50.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.52.50.png\" alt=\"valem\">\u00a0st leia valitsust toetavate isikute arv valimis,<br>3. J\u00e4reljaotuse parameetrite leidmiseks arvuta:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"154\" height=\"60\" class=\"wp-image-317 aligncenter\" title=\"screen_shot_2020-04-06_at_09.53.42.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.53.42.png\" alt=\"valem\"><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Vaatame l\u00e4hemalt nelja n\u00e4iteolukorda, kus igal juhul on k\u00fcsitletud 10 inimest (N=10), kuid varieerime eeljaotuse parameetreid ja valitsust toetavate isikute arvu.\u00a0<\/span><\/span><\/p>\n<p><i>N\u00e4ide 1<\/i>. Uudistest loetu ja sisetunde p\u00f5hjal arvab uuringu l\u00e4biviija, et valitsuse toetusm\u00e4\u00e4r on pigem madal \u2013 v\u00f5imalik, et keskmiselt umbkaudu 1 vastaja 6-st on toetaval seisukohal. Teisis\u00f5nu v\u00f5iks oodatav toetusm\u00e4\u00e4r tema hinnangul olla keskmiselt 1\/6. Kuna parameetritega\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"18\" height=\"24\" class=\"alignnone wp-image-304\" title=\"screen_shot_2020-04-06_at_09.13.21.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.21.png\" alt=\"a\">ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"19\" height=\"28\" class=\"alignnone wp-image-305\" title=\"screen_shot_2020-04-06_at_09.13.52.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.52.png\" alt=\"b\">m\u00e4\u00e4ratud beeta jaotuse keskv\u00e4\u00e4rtus on <img loading=\"lazy\" decoding=\"async\" width=\"48\" height=\"41\" class=\"alignnone wp-image-318\" title=\"screen_shot_2020-04-06_at_10.29.48.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.29.48.png\" alt=\"valem\">siis kaalub ta h\u00fcperparameetrite v\u00e4\u00e4rtustena kasutada v\u00e4\u00e4rtusi\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"134\" height=\"31\" class=\"alignnone wp-image-319\" title=\"screen_shot_2020-04-06_at_10.30.17.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.30.17.png\" alt=\"valem\">Joonistades v\u00e4lja vastava beeta jaotuse tihedusfunktsiooni, n\u00e4eb ta, et selliste parameetrite korral on liialt suur v\u00f5imalus saada nullile v\u00e4ga l\u00e4hedasi v\u00e4\u00e4rtusi (joonis 1) ning kuigi arvatavasti on toetusm\u00e4\u00e4r pigem madal, on valitsusel siiski oma toetajaskond kindlasti olemas. Kuna eelnevalt fikseeritud h\u00fcperparameetrite v\u00e4\u00e4rtused ei sobi tema eelteadmistega, otsustab uuringu l\u00e4biviija j\u00e4rgmisena proovida v\u00e4\u00e4rtusi <img loading=\"lazy\" decoding=\"async\" width=\"144\" height=\"32\" class=\"alignnone wp-image-320\" title=\"screen_shot_2020-04-06_at_10.30.51.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.30.51.png\" alt=\"valem\">Ka sel juhul on keskv\u00e4\u00e4rtus samuti 1\/6, kuid n\u00fc\u00fcd saadav tihedusfunktsiooni kuju sobib tema eelteadmistega paremini (joonis 1). Olles kaalunud ka m\u00f5ningaid teisi <img loading=\"lazy\" decoding=\"async\" width=\"18\" height=\"24\" class=\"alignnone wp-image-304\" title=\"screen_shot_2020-04-06_at_09.13.21.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.21.png\" alt=\"a\">ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"19\" height=\"28\" class=\"alignnone wp-image-305\" title=\"screen_shot_2020-04-06_at_09.13.52.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_09.13.52.png\" alt=\"b\">v\u00e4\u00e4rtusi, otsustab ta siiski j\u00e4\u00e4da algv\u00e4\u00e4rtuste\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"142\" height=\"30\" class=\"alignnone wp-image-321\" title=\"screen_shot_2020-04-06_at_10.31.58.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.31.58.png\" alt=\"valem\">\u00a0juurde.<\/p>\n<figure id=\"attachment_322\" aria-describedby=\"caption-attachment-322\" style=\"width: 563px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-322\" title=\"Joonis 1. Beeta jaotuse tihedusfunktsioonide v\u00f5rdlus parameetrite \u03b1=1 ja \u03b2=5 (punane joon) ning \u03b1=2 ja \u03b2=10 (roheline joon) korral\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.32.26.png\" alt=\"Joonis 1. Beeta jaotuse tihedusfunktsioonide v\u00f5rdlus parameetrite \u03b1=1 ja \u03b2=5 (punane joon) ning \u03b1=2 ja \u03b2=10 (roheline joon) korral\" width=\"563\" height=\"302\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.32.26.png 563w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.32.26-300x161.png 300w\" sizes=\"auto, (max-width: 563px) 100vw, 563px\"><figcaption id=\"caption-attachment-322\" class=\"wp-caption-text\">Joonis 1. Beeta jaotuse tihedusfunktsioonide v\u00f5rdlus parameetrite \u03b1=1 ja \u03b2=5 (punane joon) ning \u03b1=2 ja \u03b2=10 (roheline joon) korral<\/figcaption><\/figure>\n<p>Olles formuleerinud oma eelteadmised, asub anal\u00fc\u00fctik l\u00e4bi viima k\u00fcsitlust. Selleks k\u00fcsitleb ta 10 juhuslikult valitud isikut, kellest 6 on valitsust toetaval seisukohal <img loading=\"lazy\" decoding=\"async\" width=\"191\" height=\"31\" class=\"alignnone wp-image-323\" title=\"screen_shot_2020-04-06_at_10.33.10.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.33.10.png\" alt=\"valem\"><\/p>\n<p>Viimase sammuna tuleb veel arvutada j\u00e4reljaotuse parameetrid: <img loading=\"lazy\" decoding=\"async\" width=\"347\" height=\"35\" class=\"alignnone wp-image-324\" title=\"screen_shot_2020-04-06_at_10.33.47.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.33.47.png\" alt=\"valem\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.33.47.png 347w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.33.47-300x30.png 300w\" sizes=\"auto, (max-width: 347px) 100vw, 347px\">\u00a0<!-- [if gte msEquation 12]&gt;--><i><span lang=\"ET\" style=\"font-size: 12.0pt; line-height: 107%; font-family: 'Cambria Math',serif;\">.<\/span><\/i> \u00a0Raporteerides punkthinnanguna j\u00e4reljaotuse keskv\u00e4\u00e4rtust, on valitsuse toetusm\u00e4\u00e4ra hinnanguks <img loading=\"lazy\" decoding=\"async\" width=\"145\" height=\"45\" class=\"alignnone wp-image-325\" title=\"screen_shot_2020-04-06_at_10.34.19.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.34.19.png\" alt=\"valem\"><\/p>\n<p><i>N\u00e4ide 2. <\/i>Teisel anal\u00fc\u00fctikul on kavas viia l\u00e4bi samasugune uuring nagu n\u00e4ites 1. Sel korral on uuringu l\u00e4biviija aga ise valitsust toetav ning k\u00f5ik tema peamised tuttavad on samasugusel seisukohal. Selline kajakambris viibimine on temas tekitanud arvamuse, et valitsuse toetusm\u00e4\u00e4r \u00fchiskonnas ongi \u00fcsna k\u00f5rge. Sellest tulenevalt valib ta h\u00fcperparameetriteks\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"142\" height=\"29\" class=\"alignnone wp-image-326\" title=\"screen_shot_2020-04-06_at_10.35.24.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.35.24.png\" alt=\"valem\">\u00a0st tema ootuste kohaselt v\u00f5iks 12 isikust 10 olla valitsust toetavad. Viies l\u00e4bi k\u00fcsitluse ning saades tulemuseks, et 10 vastajast 6 toetab praegust valitsust, uuendab anal\u00fc\u00fctik oma teadmisi vastavalt arvutusalgoritmile:\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"342\" height=\"32\" class=\"alignnone wp-image-327\" title=\"screen_shot_2020-04-06_at_10.36.16.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.36.16.png\" alt=\"valem\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.36.16.png 342w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_10.36.16-300x28.png 300w\" sizes=\"auto, (max-width: 342px) 100vw, 342px\">\u00a0Toetusm\u00e4\u00e4ra punkthinnanguks on seega\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"44\" class=\"alignnone wp-image-328\" title=\"screen_shot_2020-04-06_at_11.00.58.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.00.58.png\" alt=\"valem\"><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">N\u00e4idete 1 ja 2 p\u00f5hjal on n\u00e4ha, et s\u00f5ltuvalt eelteadmistest v\u00f5ivad anal\u00fc\u00fcsi tulemused isegi samade andmete korral tugevalt erineda. Kuigi teatud juhtudel v\u00f5imaldab olemasolevate teadmiste kasutuselev\u00f5tt v\u00e4hemalt teoreetiliselt t\u00e4psemaid tulemusi saada, toob see paratamatult sisse ka subjektiivsuse komponendi. Subjektiivsus on aga tegur, mida teaduslikus uuringus v\u00f5imalikult suurel m\u00e4\u00e4ral v\u00e4ltida tuleb. Tugineda tuleb faktidele, milleks on eelk\u00f5ige andmed. <\/span><\/span><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">\u00d5nneks on Bayesi anal\u00fc\u00fcsis subjektiivsuse v\u00e4ltimiseks olemas sobiv lahendus: eeljaotuse h\u00fcperparameetriteks tuleb valida sellised v\u00e4\u00e4rtused, mille korral k\u00f5ikv\u00f5imalikud uuritava parameetri v\u00e4\u00e4rtused on eelteadmiste j\u00e4rgi v\u00f5rdv\u00f5imalikud. Selliselt m\u00e4\u00e4ratud eeljaotust nimetatakse mitteinformatiivseks eeljaotuseks (<i>non-informative prior<\/i>). Kasutades mitteinformatiivset eeljaotust, on tulemused kas samad v\u00f5i v\u00e4hemasti l\u00e4hedased klassikalise statistika meetoditega saadud tulemustele.<\/span><\/span><\/p>\n<p>N\u00e4iteks, kui beeta jaotuse korral valida h\u00fcperparameetriteks <img loading=\"lazy\" decoding=\"async\" width=\"135\" height=\"38\" class=\"alignnone wp-image-329\" title=\"screen_shot_2020-04-06_at_11.08.11.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.08.11.png\" alt=\"valem\">\u00a0, saame vahemikule 0 kuni 1 vastava \u00fchtlase jaotuse. Kui aga mingi uuritava parameetri <img loading=\"lazy\" decoding=\"async\" width=\"19\" height=\"25\" class=\"alignnone wp-image-330\" title=\"screen_shot_2020-04-06_at_11.09.45.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.09.45.png\" alt=\"screen_shot_2020-04-06_at_11.09.45.png\">eeljaotuseks on valitud normaaljaotus h\u00fcperparameetritega\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"187\" height=\"28\" class=\"alignnone wp-image-331\" title=\"screen_shot_2020-04-06_at_11.10.20.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.10.20.png\" alt=\"valem\">siis oma ebakindluse v\u00e4ljendamiseks ning subjektiivsuse v\u00e4hendamiseks saab suurendada hajuvusparameetrit <img loading=\"lazy\" decoding=\"async\" width=\"23\" height=\"23\" class=\"alignnone wp-image-332\" title=\"screen_shot_2020-04-06_at_11.10.51.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.10.51.png\" alt=\"y\"><\/p>\n<p><i>N\u00e4ide 3.<\/i> Viime l\u00e4bi n\u00e4idetes 1 ja 2 esitatud uuringu, kuid kasutame sel korral mitteinformatiivset eeljaotust. Fikseerime seega <img loading=\"lazy\" decoding=\"async\" width=\"133\" height=\"29\" class=\"alignnone wp-image-333\" title=\"screen_shot_2020-04-06_at_11.11.34.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.11.34.png\" alt=\"valem\">\u00a0Eeljaotuse kohaselt on oodatav toetusm\u00e4\u00e4r\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"90\" height=\"44\" class=\"alignnone wp-image-334\" title=\"screen_shot_2020-04-06_at_11.12.20.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.12.20.png\" alt=\"valem\">\u00a0Kuna andmete p\u00f5hjal <img loading=\"lazy\" decoding=\"async\" width=\"151\" height=\"33\" class=\"alignnone wp-image-335\" title=\"screen_shot_2020-04-06_at_11.12.53.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.12.53.png\" alt=\"valem\">\u00a0siis j\u00e4reljaotuse parameetrid on\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"324\" height=\"35\" class=\"alignnone wp-image-336\" title=\"screen_shot_2020-04-06_at_11.13.28.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.13.28.png\" alt=\"valem\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.13.28.png 324w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.13.28-300x32.png 300w\" sizes=\"auto, (max-width: 324px) 100vw, 324px\">millest omakorda saame, et <img loading=\"lazy\" decoding=\"async\" width=\"142\" height=\"42\" class=\"alignnone wp-image-337\" title=\"screen_shot_2020-04-06_at_11.15.20.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.15.20.png\" alt=\"valem\">N\u00e4eme, et saadud hinnang ei kattu k\u00fcll t\u00e4ielikult klassikalise statistika hinnanguga, milleks on \u00a0<img loading=\"lazy\" decoding=\"async\" width=\"208\" height=\"59\" class=\"alignnone wp-image-338\" title=\"screen_shot_2020-04-06_at_11.15.54.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_11.15.54.png\" alt=\"valem\"><span lang=\"ET\"><span style=\"line-height: 107%;\">kuid on sellele v\u00f5rdlemisi l\u00e4hedal.<\/span><\/span><\/p>\n<p><i>N\u00e4ide 4.<\/i> Vaatleme n\u00fc\u00fcd aga olukorda, kui 10 k\u00fcsitletust mitte \u00fckski pole valitsust toetaval seisukohal, st <img loading=\"lazy\" decoding=\"async\" width=\"66\" height=\"32\" class=\"alignnone wp-image-339\" title=\"screen_shot_2020-04-06_at_14.16.37.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.16.37.png\" alt=\"nt\">\u00a0On selge, et klassikalise statistikaga saaksime toetuse hinnanguks <img loading=\"lazy\" decoding=\"async\" width=\"149\" height=\"35\" class=\"alignnone wp-image-340\" title=\"screen_shot_2020-04-06_at_14.17.28.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.17.28.png\" alt=\"valem\">\u00a0Kuna valimi maht on v\u00e4ga v\u00e4ike, on t\u00e4iesti reaalne saada selline valim, kus k\u00f5ik on \u201evastu\u201c. K\u00fcll aga on raske aktsepteerida hinnangut, et kogu \u00fchiskonnas v\u00f5iks toetusm\u00e4\u00e4r 0 olla. Bayesi anal\u00fc\u00fcsi tulemused on seevastu m\u00f5nev\u00f5rra paindlikumad: valides eeljaotuseks mitteinformatiivse beeta jaotuse (st\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"128\" height=\"32\" class=\"alignnone wp-image-341\" title=\"screen_shot_2020-04-06_at_14.18.08.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.18.08.png\" alt=\"valem\">),\u00a0saame, et <img loading=\"lazy\" decoding=\"async\" width=\"528\" height=\"54\" class=\"alignnone wp-image-342\" title=\"screen_shot_2020-04-06_at_14.19.05.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.19.05.png\" alt=\"valem\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.19.05.png 528w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.19.05-300x31.png 300w\" sizes=\"auto, (max-width: 528px) 100vw, 528px\"><\/p>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Nagu just veenduda v\u00f5isime, <\/span><\/span><span lang=\"ET\"><span style=\"line-height: 107%;\">sobib Bayesi statistikat kasutada harva esinevate s\u00fcndmuste korral.<\/span><\/span><\/p>\n<p>Joonis 2 aitab visualiseerida, kuidas n\u00e4idetes 1 kuni 4 fikseeritud eeljaotustest ja saadud andmetest s\u00f5ltuvalt kujuneb j\u00e4reljaotus. N\u00e4idetes 1 kuni 3 on olnud kasutada samad andmed, kuid s\u00f5ltuvalt eeljaotuse valikust on j\u00e4reljaotused ja sellest tuletatavad tulemused v\u00f5rdlemisi erinevad. N\u00e4iteid 3 ja 4 v\u00f5rreldes on aga n\u00e4ha, kuidas sama (mitteinformatiivse) eeljaotuse korral on erinevus andmetes tinginud tugevad erinevused j\u00e4reljaotuses.<\/p>\n<figure id=\"attachment_343\" aria-describedby=\"caption-attachment-343\" style=\"width: 838px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-343\" title=\"Joonis 2. N\u00e4idetes 1 kuni 4 esitatud eel- ja j\u00e4reljaotuste v\u00f5rdlus N = 10 korral\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.20.03_01.png\" alt=\"Joonis 2. N\u00e4idetes 1 kuni 4 esitatud eel- ja j\u00e4reljaotuste v\u00f5rdlus N = 10 korral\" width=\"838\" height=\"437\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.20.03_01.png 838w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.20.03_01-300x156.png 300w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.20.03_01-768x400.png 768w\" sizes=\"auto, (max-width: 838px) 100vw, 838px\"><figcaption id=\"caption-attachment-343\" class=\"wp-caption-text\">Joonis 2. N\u00e4idetes 1 kuni 4 esitatud eel- ja j\u00e4reljaotuste v\u00f5rdlus N = 10 korral<\/figcaption><\/figure>\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">N\u00e4idetes 1 kuni 4 olev sisu on kokku koondatud tabelis 1.<\/span><\/span><\/p>\n<p><em>Tabel 1. <\/em>N\u00e4idetes 1 kuni 4 leitud punkthinnangute v\u00f5rdlus N = 10 korral<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"797\" height=\"186\" class=\"alignnone wp-image-344\" title=\"Tabel 1. N\u00e4idetes 1 kuni 4 leitud punkthinnangute v\u00f5rdlus N = 10 korral\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.21.16.png\" alt=\"Tabel 1. N\u00e4idetes 1 kuni 4 leitud punkthinnangute v\u00f5rdlus N = 10 korral\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.21.16.png 797w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.21.16-300x70.png 300w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.21.16-768x179.png 768w\" sizes=\"auto, (max-width: 797px) 100vw, 797px\"><\/p>\n<p>On selge, et valim mahuga 10 on v\u00e4ga v\u00e4ike. Mis toimub aga valimi mahu kasvades? Intuitiivselt v\u00f5iks oodata, et mida enam on andmeid (st mida k\u00f5nekamad on faktid), seda tugevamalt v\u00f5tame neid arvesse oma j\u00e4relduste tegemisel. Teisis\u00f5nu v\u00f5iks eelteadmiste osat\u00e4htsus t\u00e4iendavate faktide ilmnemisel aina v\u00e4heneda. Osutub, et Bayesi statistikas see nii ka on.<\/p>\n<p>Vaatleme n\u00e4idetes 1 kuni 4 esitatud olukordi, kuid v\u00f5tame n\u00fc\u00fcd valimi mahuks <img loading=\"lazy\" decoding=\"async\" width=\"100\" height=\"30\" class=\"alignnone wp-image-345\" title=\"screen_shot_2020-04-06_at_14.22.41.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.22.41.png\" alt=\"n\">j\u00e4ttes valitsust toetavate isikute valimisisese proportsiooni samaks, st n\u00e4idetes 1 kuni 3\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"94\" height=\"38\" class=\"alignnone wp-image-346\" title=\"screen_shot_2020-04-06_at_14.23.20.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.23.20.png\" alt=\"Nt\">ja n\u00e4ites 4\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"60\" height=\"35\" class=\"alignnone wp-image-347\" title=\"screen_shot_2020-04-06_at_14.23.58.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.23.58.png\" alt=\"Nt\">.\u00a0Jooniselt 3 on n\u00e4ha, kuidas toetusm\u00e4\u00e4ra <img loading=\"lazy\" decoding=\"async\" width=\"21\" height=\"27\" class=\"alignnone wp-image-348\" title=\"screen_shot_2020-04-06_at_14.24.35.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.24.35.png\" alt=\"O\">j\u00e4reljaotus on tugevalt kontsentreeritud valimisisese proportsiooni \u00fcmbrusesse, ning kuigi n\u00e4idetes 1 kuni 3 esitatud eeljaotused on t\u00e4iesti erinevad, siis j\u00e4reljaotused on visuaalselt v\u00e4ga sarnased.<\/p>\n<figure id=\"attachment_349\" aria-describedby=\"caption-attachment-349\" style=\"width: 835px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-349\" title=\"Joonis 3. N\u00e4idetes 1 kuni 4 esitatud eel- ja j\u00e4reljaotuste v\u00f5rdlus N = 1000 korral\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.25.08.png\" alt=\"Joonis 3. N\u00e4idetes 1 kuni 4 esitatud eel- ja j\u00e4reljaotuste v\u00f5rdlus N = 1000 korral\" width=\"835\" height=\"421\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.25.08.png 835w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.25.08-300x151.png 300w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.25.08-768x387.png 768w\" sizes=\"auto, (max-width: 835px) 100vw, 835px\"><figcaption id=\"caption-attachment-349\" class=\"wp-caption-text\">Joonis 3. N\u00e4idetes 1 kuni 4 esitatud eel- ja j\u00e4reljaotuste v\u00f5rdlus N = 1000 korral<\/figcaption><\/figure>\n<p>\u00dchtlasi saame tabeli 2 p\u00f5hjal veenduda, et Bayesi statistika p\u00f5hjal leitud punkthinnangud (<img loading=\"lazy\" decoding=\"async\" width=\"93\" height=\"33\" class=\"alignnone wp-image-350\" title=\"screen_shot_2020-04-06_at_14.40.45.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.40.45.png\" alt=\"valem\">)\u00a0<span lang=\"ET\"><span style=\"line-height: 107%;\">on juba v\u00e4ga l\u00e4hedased klassikalise statistika hinnangutele.<\/span><\/span><\/p>\n<p><em>Tabel 2. <\/em>N\u00e4idetes 1 kuni 4 leitud punkthinnangute v\u00f5rdlus N = 1000 korral<img loading=\"lazy\" decoding=\"async\" width=\"858\" height=\"180\" class=\"alignnone wp-image-351\" title=\"Tabel 2. N\u00e4idetes 1 kuni 4 leitud punkthinnangute v\u00f5rdlus N = 1000 korral\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.41.31.png\" alt=\"Tabel 2. N\u00e4idetes 1 kuni 4 leitud punkthinnangute v\u00f5rdlus N = 1000 korral\" srcset=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.41.31.png 858w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.41.31-300x63.png 300w, https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.41.31-768x161.png 768w\" sizes=\"auto, (max-width: 858px) 100vw, 858px\"><\/p>\n<p><\/p><\/div>\n        <\/div>\n        <\/div>\n    <\/div>\n<p><\/p><div class=\"accordion mb-3\">\n        <div class=\"accordion-item accordion-item--white\">\n        <h2 class=\"accordion-header\" id=\"accordion-69de8869e1d82-heading\">\n            <button class=\"accordion-button collapsed\" type=\"button\" data-bs-toggle=\"collapse\" data-bs-target=\"#accordion-69de8869e1d82-collapse\" aria-expanded=\"true\" aria-controls=\"accordion-69de8869e1d82-collapse\">T\u00f5en\u00e4osusintervall<\/button>\n        <\/h2>\n        <div id=\"accordion-69de8869e1d82-collapse\" class=\"accordion-collapse collapse\" aria-labelledby=\"accordion-69de8869e1d82-heading\">\n            <div class=\"accordion-body\">\n<p>Sagedusstatistikud kasutavad uuritava parameetri punkthinnangu usaldusv\u00e4\u00e4rsuse kirjeldamiseks <a title=\"\" href=\"http:\/\/samm.ut.ee\/usaldusvahemik\" target=\"_blank\" rel=\"noopener\" data-url=\"http:\/\/samm.ut.ee\/usaldusvahemik\">usaldusvahemikku<\/a>.\u00a0<span lang=\"ET\"><span style=\"line-height: 107%;\">Klassikalise statistika eelduse kohaselt on uuritav parameeter fikseeritud ning juhuslik on vaid saadav valim. Seega on juhuslik ka valimi p\u00f5hjal arvutatav usaldusvahemik, mist\u00f5ttu on v\u00e4\u00e4r n\u00e4iteks 95% usaldusvahemikku t\u00f5lgendada selliselt, et <i>t\u00f5en\u00e4osusega 95% kuulub tegelik \u00fcldkogumi parameeter leitud vahemikku<\/i>. Kuna nii leitud usaldusvahemik kui ka tegelik parameeter on fikseeritud, siis tegelik parameeter kas kuulub v\u00f5i ei kuulu leitud vahemikku, st t\u00f5en\u00e4osus, et \u00fcldkogumi parameeter kuulub leitud vahemikku, on vastavalt kas 1 v\u00f5i 0. Korrektne 95% usaldusvahemiku t\u00f5lgendus on hoopis selline: kui korrata valimi v\u00f5tmise protsessi ja iga valimi korral arvutada 95% usaldusvahemik, siis oodatavalt 95% juhtudest j\u00e4\u00e4b tegelik parameeter leitud vahemikku.\u00a0<\/span><\/span><\/p>\n<p>Bayesi statistikud kasutavad usaldusvahemiku asemel t\u00f5en\u00e4osusvahemikku (<i>probability interval, credibility interval<\/i>). Kuna Bayesi statistika l\u00e4htub eeldusest, et \u00fcldkogumi parameeter on juhuslik suurus, siis saame r\u00e4\u00e4kida t\u00f5en\u00e4osusest, et see kuulub mingisse etteantud vahemikku. N\u00e4iteks parameetri <img loading=\"lazy\" decoding=\"async\" width=\"60\" height=\"33\" class=\"alignnone wp-image-352\" title=\"screen_shot_2020-04-06_at_14.44.05.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.44.05.png\" alt=\"valem\">t\u00f5en\u00e4osusvahemik\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"51\" height=\"30\" class=\"alignnone wp-image-353\" title=\"screen_shot_2020-04-06_at_14.44.36.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.44.36.png\" alt=\"T\"><span lang=\"ET\"><span style=\"line-height: 107%;\">defineeritakse kui <\/span><\/span><img loading=\"lazy\" decoding=\"async\" width=\"199\" height=\"43\" class=\"alignnone wp-image-354\" title=\"screen_shot_2020-04-06_at_14.45.06.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.45.06.png\" alt=\"valem\">Seega on t\u00f5en\u00e4osusvahemiku t\u00f5lgendamine tunduvalt intuitiivsem kui usaldusvahemiku t\u00f5lgendamine: n\u00e4iteks 95% t\u00f5en\u00e4osusvahemiku korral on t\u00f5lgenduseks \u201et\u00f5en\u00e4osusega 95% kuulub \u00fcldkogumi parameeter vahemikku <img loading=\"lazy\" decoding=\"async\" width=\"60\" height=\"31\" class=\"alignnone wp-image-355\" title=\"screen_shot_2020-04-06_at_14.45.40.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.45.40.png\" alt=\"T\">.<\/p>\n<p>T\u00f5en\u00e4osusvahemiku leidmiseks tuleb p\u00f6\u00f6rata t\u00e4helepanu asjaolule, et see ei ole \u00fcldiselt \u00fcheselt m\u00e4\u00e4ratud. N\u00e4iteks sobivad 95% t\u00f5en\u00e4osusvahemikuks nii (\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"82\" height=\"33\" class=\"alignnone wp-image-356\" title=\"screen_shot_2020-04-06_at_14.46.28.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.46.28.png\" alt=\"valem\">) kui ka (<img loading=\"lazy\" decoding=\"async\" width=\"69\" height=\"29\" class=\"alignnone wp-image-357\" title=\"screen_shot_2020-04-06_at_14.47.14.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.47.14.png\" alt=\"valem\">), kus\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"44\" height=\"33\" class=\"alignnone wp-image-358\" title=\"screen_shot_2020-04-06_at_14.48.42.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.48.42.png\" alt=\"q\">\u00a0ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"48\" height=\"29\" class=\"alignnone wp-image-359\" title=\"screen_shot_2020-04-06_at_14.49.01.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.49.01.png\" alt=\"q\">on vastavalt uuritava parameetri <img loading=\"lazy\" decoding=\"async\" width=\"17\" height=\"27\" class=\"alignnone wp-image-360\" title=\"screen_shot_2020-04-06_at_14.49.26.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.49.26.png\" alt=\"screen_shot_2020-04-06_at_14.49.26.png\">j\u00e4reljaotuse 0,95 ja 0,05 kvantiilid. \u00dcldiselt on siiski m\u00f5istlik valida v\u00f5imalikest t\u00f5en\u00e4osusvahemikest k\u00f5ige l\u00fchema pikkusega vahemik v\u00f5i selline vahemik, mille saamiseks on jaotuse m\u00f5lemast sabast \u00e4ra visatud sama suur t\u00f5en\u00e4osusmass: kui on teada j\u00e4reljaotuse anal\u00fc\u00fctiline kuju, saab leida 0,025 ja 0,975 kvantiilid <img loading=\"lazy\" decoding=\"async\" width=\"52\" height=\"31\" class=\"alignnone wp-image-361\" title=\"screen_shot_2020-04-06_at_14.50.08.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.50.08.png\" alt=\"q\">ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"56\" height=\"31\" class=\"alignnone wp-image-362\" title=\"screen_shot_2020-04-06_at_14.50.57.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.50.57.png\" alt=\"q\">ning nendest konstrueeritud vahemik (<img loading=\"lazy\" decoding=\"async\" width=\"104\" height=\"28\" class=\"alignnone wp-image-363\" title=\"screen_shot_2020-04-06_at_14.51.27.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.51.27.png\" alt=\"qq\">)\u00a0ongi soovitud 95% t\u00f5en\u00e4osusvahemik. Kui anal\u00fc\u00fctiliselt ei \u00f5nnestu soovitud kvantiile arvutada, on neid v\u00f5imalik hinnata j\u00e4reljaotusest simuleerimise teel. Selleks tuleb j\u00e4reljaotuse p\u00f5hjal genereerida <img loading=\"lazy\" decoding=\"async\" width=\"48\" height=\"35\" class=\"alignnone wp-image-364\" title=\"screen_shot_2020-04-06_at_14.52.07.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_14.52.07.png\" alt=\"N\">v\u00f5imalikku realisatsiooni ning j\u00e4rjestada need kasvavalt. Kvantiili <img loading=\"lazy\" decoding=\"async\" width=\"27\" height=\"29\" class=\"alignnone wp-image-365\" title=\"screen_shot_2020-04-06_at_15.01.48.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.01.48.png\" alt=\"q\">\u00a0hinnanguks on positsioonil\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"70\" height=\"33\" class=\"alignnone wp-image-366\" title=\"screen_shot_2020-04-06_at_15.02.24.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.02.24.png\" alt=\"q\">asuv element (seda juhul, kui <img loading=\"lazy\" decoding=\"async\" width=\"71\" height=\"27\" class=\"alignnone wp-image-367\" title=\"screen_shot_2020-04-06_at_15.03.03.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.03.03.png\" alt=\"Nsim\">on t\u00e4isarv, vastasel juhul tuleb t\u00f5en\u00e4osusvahemiku vasaku otspunkti leidmiseks suurust <img loading=\"lazy\" decoding=\"async\" width=\"71\" height=\"27\" class=\"alignnone wp-image-367\" title=\"screen_shot_2020-04-06_at_15.03.03.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.03.03.png\" alt=\"Nsim\">\u00fcmardada l\u00e4hima t\u00e4isarvuni allapoole ja parema otspunkti leidmiseks \u00fcmardada l\u00e4hima t\u00e4isarvuni \u00fclespoole). N\u00e4ites\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"113\" height=\"34\" class=\"alignnone wp-image-368\" title=\"screen_shot_2020-04-06_at_15.04.19.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.04.19.png\" alt=\"N\">korral tuleb 95% t\u00f5en\u00e4osusvahemiku leidmiseks valida kasvavalt j\u00e4rjestatud simuleeritud v\u00e4\u00e4rtustest positsioonidel\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"159\" height=\"34\" class=\"alignnone wp-image-369\" title=\"screen_shot_2020-04-06_at_15.06.11.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.06.11.png\" alt=\"valem\">ja\u00a0<img loading=\"lazy\" decoding=\"async\" width=\"173\" height=\"41\" class=\"alignnone wp-image-370\" title=\"screen_shot_2020-04-06_at_15.06.32.png\" src=\"https:\/\/sisu.ut.ee\/wp-content\/uploads\/sites\/110\/screen_shot_2020-04-06_at_15.06.32.png\" alt=\"valem\">asuvad v\u00e4\u00e4rtused.<\/p>\n<p><\/p><\/div>\n        <\/div>\n        <\/div>\n    <\/div>\n<p><\/p><div class=\"accordion mb-3\">\n        <div class=\"accordion-item accordion-item--white\">\n        <h2 class=\"accordion-header\" id=\"accordion-69de8869e1d87-heading\">\n            <button class=\"accordion-button collapsed\" type=\"button\" data-bs-toggle=\"collapse\" data-bs-target=\"#accordion-69de8869e1d87-collapse\" aria-expanded=\"true\" aria-controls=\"accordion-69de8869e1d87-collapse\">J\u00e4reljaotuse simuleerimine<\/button>\n        <\/h2>\n        <div id=\"accordion-69de8869e1d87-collapse\" class=\"accordion-collapse collapse\" aria-labelledby=\"accordion-69de8869e1d87-heading\">\n            <div class=\"accordion-body\">\n<p><span lang=\"ET\"><span style=\"line-height: 107%;\">Tihtipeale ei ole j\u00e4reljaotust v\u00f5imalik anal\u00fc\u00fctiliselt leida. Sel juhul tuleb kasutada simuleerimist. Levinud l\u00fchendiks on MCMC (<i>Markov Chain Monte Carlo<\/i>)<i>, <\/i>mis kujutab endast Markovi ahelatel p\u00f5hinevaid simuleerimismeetodeid. Nendest algoritmidest \u00fclevaate saamiseks on soovitatav tutvuda n\u00e4iteks alloleva kirjandusega.<\/span><\/span><\/p>\n<p><\/p><\/div>\n        <\/div>\n        <\/div>\n    <\/div>\n<p><\/p><div class=\"accordion mb-3\">\n        <div class=\"accordion-item accordion-item--white\">\n        <h2 class=\"accordion-header\" id=\"accordion-69de8869e1d8a-heading\">\n            <button class=\"accordion-button collapsed\" type=\"button\" data-bs-toggle=\"collapse\" data-bs-target=\"#accordion-69de8869e1d8a-collapse\" aria-expanded=\"true\" aria-controls=\"accordion-69de8869e1d8a-collapse\">Soovitatav kirjandus<\/button>\n        <\/h2>\n        <div id=\"accordion-69de8869e1d8a-collapse\" class=\"accordion-collapse collapse\" aria-labelledby=\"accordion-69de8869e1d8a-heading\">\n            <div class=\"accordion-body\">\n<p><span style=\"line-height: normal;\"><span lang=\"EN-US\"><span style=\"letter-spacing: .1pt;\">Clyde, M. et al. (2019). <i>An Introduction to Bayesian Thinking. <\/i><\/span><\/span><span lang=\"DE\">K\u00e4ttesaadav internetist aadressil<\/span><i> <\/i><a href=\"https:\/\/statswithr.github.io\/book\/\"><span lang=\"DE\">https:\/\/statswithr.github.io\/book\/<\/span><\/a><span lang=\"ET\">.<\/span><\/span><\/p>\n<p><span style=\"line-height: normal;\"><span lang=\"EN-US\"><span style=\"letter-spacing: .1pt;\">Koch, <\/span><\/span><span lang=\"EN-US\">K.-R. <\/span><span lang=\"EN-US\"><span style=\"letter-spacing: .1pt;\">(2007). <i>Introduction to Bayesian Statistics<\/i>. <\/span><\/span><span lang=\"DE\"><span style=\"letter-spacing: .1pt;\">Berlin, Heidelberg: Springer.<\/span><\/span><\/span><\/p>\n<p><span style=\"line-height: normal;\"><span lang=\"ET\">Lync, <\/span><span lang=\"EN-US\">S. M. (2007). <i>Introduction to Applied Bayesian Statistics and Estimation for Social Scientists<\/i>. New York: Springer.<\/span><\/span><\/p>\n<p><span style=\"line-height: normal;\"><span lang=\"DE\">J. Esarey videokursus:\u00a0 <\/span><a href=\"https:\/\/www.youtube.com\/playlist?list=PLAFC5F02F224FA59F\"><span lang=\"DE\">https:\/\/www.youtube.com\/playlist?list=PLAFC5F02F224FA59F<\/span><\/a><\/span><\/p>\n<p><\/p><\/div>\n        <\/div>\n        <\/div>\n    <\/div>\n<p><i>Valminud Hariduse Infotehnoloogia Sihtasutuse IT Akadeemia programmi toel<\/i>.<\/p>\n<p style=\"margin: 0cm0cm8pt;\">\u00a0<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Taavi Unt2020 Statistikas on eristatavad kaks peamist koolkonda: Bayesi statistika ja sagedusstatistika ehk nn klassikaline statistika.\u00a0M\u00f5lemal juhul on eesm\u00e4rk olemasolevate vaatluste (valimi) pealt hinnata kogu populatsioonis kehtivaid seoseid v\u00f5i huvipakkuvaid n\u00e4itajaid. Teisis\u00f5nu soovitakse hinnata parameetrit \u03b8\u00a0(theta), mis v\u00f5ib olla nii &#8230;<\/p>\n","protected":false},"author":45,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"class_list":["post-39","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sisu.ut.ee\/samm\/wp-json\/wp\/v2\/pages\/39","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sisu.ut.ee\/samm\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sisu.ut.ee\/samm\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sisu.ut.ee\/samm\/wp-json\/wp\/v2\/users\/45"}],"replies":[{"embeddable":true,"href":"https:\/\/sisu.ut.ee\/samm\/wp-json\/wp\/v2\/comments?post=39"}],"version-history":[{"count":22,"href":"https:\/\/sisu.ut.ee\/samm\/wp-json\/wp\/v2\/pages\/39\/revisions"}],"predecessor-version":[{"id":2257,"href":"https:\/\/sisu.ut.ee\/samm\/wp-json\/wp\/v2\/pages\/39\/revisions\/2257"}],"wp:attachment":[{"href":"https:\/\/sisu.ut.ee\/samm\/wp-json\/wp\/v2\/media?parent=39"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}