{"id":5129,"date":"2021-06-14T11:33:53","date_gmt":"2021-06-14T09:33:53","guid":{"rendered":"https:\/\/dev.m2hycon.de\/aktuelles\/unkategorisiert\/bayesian-statistics-as-an-extension-of-machine-learning-methods\/"},"modified":"2024-04-22T12:09:29","modified_gmt":"2024-04-22T10:09:29","slug":"bayesian-statistics-as-an-extension-of-machine-learning-methods","status":"publish","type":"post","link":"https:\/\/www.m2hycon.de\/en\/news\/wiki\/bayesian-statistics-as-an-extension-of-machine-learning-methods\/","title":{"rendered":"Bayesian statis\u00adtics as an exten\u00adsion of machine learning methods"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"5129\" class=\"elementor elementor-5129 elementor-1817\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ff85573 elementor-section-full_width elementor-section-height-min-height red noisy elementor-section-height-default elementor-section-items-middle\" data-id=\"ff85573\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-482e998\" data-id=\"482e998\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a820fea elementor-widget elementor-widget-theme-post-featured-image elementor-widget-image\" data-id=\"a820fea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"theme-post-featured-image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"512\" src=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_bayessche_statistik_1920x960-1024x512.jpg\" class=\"attachment-large size-large wp-image-4853\" alt srcset=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_bayessche_statistik_1920x960-1024x512.jpg 1024w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_bayessche_statistik_1920x960-300x150.jpg 300w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_bayessche_statistik_1920x960-768x384.jpg 768w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_bayessche_statistik_1920x960-1536x768.jpg 1536w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_bayessche_statistik_1920x960.jpg 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\">\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8c2be4e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8c2be4e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-923e421\" data-id=\"923e421\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7118ddc elementor-widget elementor-widget-theme-post-title elementor-page-title elementor-widget-heading\" data-id=\"7118ddc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"theme-post-title.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Bayesian statis\u00adtics as an exten\u00adsion of machine learning methods<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b319741 elementor-widget elementor-widget-post-info\" data-id=\"b319741\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"post-info.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-inline-items elementor-icon-list-items elementor-post-info\">\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item elementor-repeater-item-0923e6a elementor-inline-item\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text elementor-post-info__item elementor-post-info__item--type-custom\">\n\t\t\t\t\t\t\t\t\t\tVon Bj\u00f6rn Piepen\u00adburg\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t<li class=\"elementor-icon-list-item elementor-repeater-item-2efc209 elementor-inline-item\" itemprop=\"datePublished\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text elementor-post-info__item elementor-post-info__item--type-date\">\n\t\t\t\t\t\t\t\t\t\t<time>June 14, 2021<\/time>\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t<li class=\"elementor-icon-list-item elementor-repeater-item-84e7708 elementor-inline-item\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text elementor-post-info__item elementor-post-info__item--type-custom\">\n\t\t\t\t\t\t\t\t\t\t<a href=\"https:\/\/www.m2hycon.de\/en\/tag\/bayesian-theorem\/\" rel=\"tag\">Bayesian theorem<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/conditional-probabilities\/\" rel=\"tag\">condi\u00adtional proba\u00adbi\u00adli\u00adties<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/posteriori-en\/\" rel=\"tag\">Poste\u00adriori<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/probabilities\/\" rel=\"tag\">Proba\u00adbi\u00adli\u00adties<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/statistics\/\" rel=\"tag\">Statis\u00adtics<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/thomas-bayes-en\/\" rel=\"tag\">Thomas Bayes<\/a>\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-f9ac5b0 elementor-section-content-middle elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f9ac5b0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-c38a12e\" data-id=\"c38a12e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5c8e1ed elementor-widget__width-auto elementor-widget elementor-widget-text-editor\" data-id=\"5c8e1ed\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h4>Share post:<\/h4>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ae685d9 elementor-share-buttons--view-icon elementor-share-buttons--skin-minimal elementor-share-buttons--color-custom elementor-widget__width-auto elementor-share-buttons--shape-circle elementor-grid-0 elementor-widget elementor-widget-share-buttons\" data-id=\"ae685d9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"share-buttons.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-grid\" role=\"list\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-grid-item\" role=\"listitem\">\n\t\t\t\t\t\t<div class=\"elementor-share-btn elementor-share-btn_facebook\" role=\"button\" tabindex=\"0\" aria-label=\"Share on facebook\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-share-btn__icon\">\n\t\t\t\t\t\t\t\t<i class=\"fab fa-facebook\" aria-hidden=\"true\"><\/i>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-grid-item\" role=\"listitem\">\n\t\t\t\t\t\t<div class=\"elementor-share-btn elementor-share-btn_twitter\" role=\"button\" tabindex=\"0\" aria-label=\"Share on twitter\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-share-btn__icon\">\n\t\t\t\t\t\t\t\t<i class=\"fab fa-twitter\" aria-hidden=\"true\"><\/i>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-grid-item\" role=\"listitem\">\n\t\t\t\t\t\t<div class=\"elementor-share-btn elementor-share-btn_linkedin\" role=\"button\" tabindex=\"0\" aria-label=\"Share on linkedin\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-share-btn__icon\">\n\t\t\t\t\t\t\t\t<i class=\"fab fa-linkedin\" aria-hidden=\"true\"><\/i>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-grid-item\" role=\"listitem\">\n\t\t\t\t\t\t<div class=\"elementor-share-btn elementor-share-btn_xing\" role=\"button\" tabindex=\"0\" aria-label=\"Share on xing\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-share-btn__icon\">\n\t\t\t\t\t\t\t\t<i class=\"fab fa-xing\" aria-hidden=\"true\"><\/i>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f408ab5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f408ab5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-281546f\" data-id=\"281546f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d986110 maths elementor-widget elementor-widget-text-editor\" data-id=\"d986110\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><em>Thomas Bayes<\/em> was born in London at the begin\u00adning of the 18th century as the son of a vicar and also became a vicar after studying theology. His other interests were logic and statis\u00adtics, which he also resear\u00adched in his spare time. His major scien\u00adtific contri\u00adbu\u00adtion is the so-called <em>Bayes\u2019 theorem<\/em>, which was only published three years after his death<\/p>\n<p>$P(A_k|E)=\\frac{P(A_k)\\cdot P(E|A_k)}{\\sum_{i=1}^k P(A_i)\\cdot P(E|A_i)}$<\/p>\n<p>As an example for the appli\u00adca\u00adtion, we take a medical rapid test, which provides a positive test result in 95% of sick people ($P(positive|sick)=0.95$). In 2% of healthy people, the test also falsely leads to a positive result ($P(positive|healthy)=0.02$). The disease has infected 2% of all people ($P(sick)=0.02$ and corre\u00adspon\u00addingly $P(healthy)=0.98$) and all people could be tested. Question: If a person tests positive, what is the proba\u00adbi\u00adlity that they actually have the disease?<\/p>\n<p>$P(sick|positive)=\\frac{P(sick)\\cdot P(positive|sick)}{P(sick)\\cdot P(positive|sick)+P(healthy)\\cdot P(positive|healthy)}=49%$<\/p>\n<p>Based on Bayes\u2019 theorem, <em>Bayesian statis\u00adtics<\/em> has developed, which is used in the context of induc\u00adtive statis\u00adtics and machine learning to estimate parame\u00adters and test hypotheses. For this purpose, the parame\u00adters are initi\u00adally assigned assumed distri\u00adbu\u00adtions (so-called <em>a priori distri\u00adbu\u00adtions<\/em>). Itera\u00adtively, the distri\u00adbu\u00adtions are adapted to the problem using statis\u00adtics from samples or the results of experi\u00adments (the a priori distri\u00adbu\u00adtions become <em>post priori distri\u00adbu\u00adtions<\/em>).<\/p>\n<p>One example that is frequently used in the litera\u00adture is the experi\u00admental deter\u00admi\u00adna\u00adtion of the proba\u00adbi\u00adlity of winning in one-armed bandits. For example, let\u2019s take three bandits with diffe\u00adrent (unknown) proba\u00adbi\u00adli\u00adties of winning (the result of a game is only a win or no win with a constant win amount). Since we have no prior knowledge, we assume a beta distri\u00adbu\u00adtion with the parame\u00adters $a=1$ and $b=1$ (corre\u00adsponds to a uniform distri\u00adbu\u00adtion) for the proba\u00adbi\u00adli\u00adties of winning. To deter\u00admine the post-priori distri\u00adbu\u00adtions, we itera\u00adtively select a bandit (depen\u00adding on the experi\u00adence already gained) and adjust its win proba\u00adbi\u00adlity curve accor\u00adding to the outcome of the game. You can cancel the proce\u00addure if the proba\u00adbi\u00adlity curves of the three bandits no longer change signi\u00adfi\u00adcantly.<\/p>\n<p>The follo\u00adwing figures show the results for the described test after 5, 10, 20, 50, 100 and 200 games. The actual win proba\u00adbi\u00adlity of the blue bandit is 0.2, that of the green bandit 0.5 and that of the red bandit 0.75. You can see the develo\u00adp\u00adments from the a priori proba\u00adbi\u00adli\u00adties (all proba\u00adbi\u00adli\u00adties of winning are equally likely) to the post priori proba\u00adbi\u00adli\u00adties.<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1fb29f2 elementor-widget elementor-widget-image\" data-id=\"1fb29f2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"768\" height=\"410\" src=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/2023\/02\/m2hycon_blog_bayessche_statistik_in_text-768x410.jpg\" class=\"attachment-medium_large size-medium_large wp-image-4975\" alt srcset=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/2023\/02\/m2hycon_blog_bayessche_statistik_in_text-768x410.jpg 768w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2023\/02\/m2hycon_blog_bayessche_statistik_in_text-300x160.jpg 300w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2023\/02\/m2hycon_blog_bayessche_statistik_in_text-1024x546.jpg 1024w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2023\/02\/m2hycon_blog_bayessche_statistik_in_text.jpg 1080w\" sizes=\"(max-width: 768px) 100vw, 768px\">\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e9ab5c7 maths elementor-widget elementor-widget-text-editor\" data-id=\"e9ab5c7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>In addition to the estimated proba\u00adbi\u00adli\u00adties of winning, the figures show the spread in the results. These can be inter\u00adpreted as certainty or uncer\u00adtainty for the assump\u00adtion of a profit proba\u00adbi\u00adlity. In order to be able to use this added value of infor\u00adma\u00adtion for diffe\u00adrent appli\u00adca\u00adtions, machine learning and artifi\u00adcial intel\u00adli\u00adgence algorithms are extended by Bayesian statis\u00adtical approa\u00adches.<\/p>\n<p>To illus\u00adtrate: Imagine you have a problem that needs to be solved on the basis of data. Experi\u00adence has shown that empirical data is subject to a certain degree of varia\u00adbi\u00adlity, is flawed, parti\u00adally incom\u00adplete and, in summary, not unambi\u00adguous. You use this data to train your model and the result is a value that appar\u00adently repres\u00adents the correct result for your problem. But how can this be if the data basis is not clear? The solution algorithm must there\u00adfore be adapted so that all data problems are taken into account in the result. To achieve this, adapted solution methods are currently being developed for relevant machine learning algorithms that take into account the scatter in the data in each calcu\u00adla\u00adtion step and output a distri\u00adbu\u00adtion as the result. One example is artifi\u00adcial neural networks, in which not only the outputs are replaced by post-priori distri\u00adbu\u00adtions, but also the network weights. In addition to the solution methods, inter\u00adfaces for further proces\u00adsing the results in the form of distri\u00adbu\u00adtions must also be adapted. For example, we used Bayesian statis\u00adtics to deter\u00admine bid prices in a research project funded by the mFund on the basis of a data base that was weak in places.<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c519333 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c519333\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-ac9195e\" data-id=\"ac9195e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b592d10 elementor-widget elementor-widget-author-box\" data-id=\"b592d10\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"author-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-author-box\">\n\t\t\t\t\t\t\t<div class=\"elementor-author-box__avatar\">\n\t\t\t\t\t<img decoding=\"async\" src=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/2023\/09\/m2hycon_ueberuns_team_Bjoern_Piepenburg_800x800-300x300.jpg\" alt=\"Picture of Bj\u00f6rn Piepenburg\" loading=\"lazy\">\n\t\t\t\t<\/div>\n\t\t\t\n\t\t\t<div class=\"elementor-author-box__text\">\n\t\t\t\t\t\t\t\t\t<div>\n\t\t\t\t\t\t<h4 class=\"elementor-author-box__name\">\n\t\t\t\t\t\t\tBj\u00f6rn Piepen\u00adburg\t\t\t\t\t\t<\/h4>\n\t\t\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element 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<\/p>\n","protected":false},"author":3,"featured_media":4853,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"wp_typography_post_enhancements_disabled":false,"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"disabled","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"disabled","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center 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